Examples with solutions for Powers of a Fraction: Applying the formula

Exercise #1

Insert the corresponding expression:

(68)2= \left(\frac{6}{8}\right)^2=

Video Solution

Step-by-Step Solution

To solve this problem, we'll compute (68)2\left(\frac{6}{8}\right)^2 using the rule for powers of a fraction:

  • Step 1: Identify the numerator and denominator: a=6a = 6 and b=8b = 8.
  • Step 2: Apply the formula (ab)n=anbn\left(\frac{a}{b}\right)^n = \frac{a^n}{b^n} with n=2n = 2.
  • Step 3: Calculate an=62a^n = 6^2.
  • Step 4: Calculate bn=82b^n = 8^2.

Now, let's carry out the calculations:

Step 3: 62=366^2 = 36.

Step 4: 82=648^2 = 64.

Therefore, (68)2=3664\left(\frac{6}{8}\right)^2 = \frac{36}{64}.

We compare 3664\frac{36}{64} with the given answer choices:

  • Choice 1: 1216\frac{12}{16} is not equal.
  • Choice 2: 664\frac{6}{64} is not equal.
  • Choice 3: 3664\frac{36}{64} matches our result.
  • Choice 4: 368\frac{36}{8} is not equal.

Therefore, the correct answer is 3664\frac{36}{64}, which matches Choice 3.

Answer

3664 \frac{36}{64}

Exercise #2

Insert the corresponding expression:

(32)3= \left(\frac{3}{2}\right)^3=

Video Solution

Step-by-Step Solution

To solve (32)3 \left(\frac{3}{2}\right)^3 , follow these steps:

  • Step 1: Identify the fraction to be raised to a power, 32\frac{3}{2}, and the exponent, 3.
  • Step 2: Apply the power to both the numerator and the denominator separately using the exponent rule for fractions.

Let's evaluate:

Raise the numerator 3 to the power of 3:

33=3×3×3=273^3 = 3 \times 3 \times 3 = 27

Raise the denominator 2 to the power of 3:

23=2×2×2=82^3 = 2 \times 2 \times 2 = 8

Combine these results into the fraction 278\frac{27}{8}.

Therefore, (32)3=278 \left(\frac{3}{2}\right)^3 = \frac{27}{8} .

Given several answer choices, the correct choice is 4: 278 \frac{27}{8} .

Answer

278 \frac{27}{8}

Exercise #3

Insert the corresponding expression:

(47)2= \left(\frac{4}{7}\right)^2=

Video Solution

Step-by-Step Solution

To solve the problem of squaring the fraction (47)2\left(\frac{4}{7}\right)^2, we will follow these steps:

  • Step 1: Determine the square of the numerator. The numerator is 44, and 42=164^2 = 16.
  • Step 2: Determine the square of the denominator. The denominator is 77, and 72=497^2 = 49.
  • Step 3: Combine these results to form the fraction 1649\frac{16}{49}.

The computation involves squaring both the numerator and the denominator separately. In conclusion, the squared fraction is:

1649 \frac{16}{49}

Therefore, the corresponding expression for (47)2\left(\frac{4}{7}\right)^2 is 1649 \frac{16}{49} .

Answer

1649 \frac{16}{49}

Exercise #4

Insert the corresponding expression:

(23)2= \left(\frac{2}{3}\right)^2=

Video Solution

Step-by-Step Solution

To solve this problem, we'll follow these steps:

  • Apply the exponentiation rule to the fraction 23\frac{2}{3}.
  • Calculate 222^2 and 323^2.
  • Form the result as a fraction.
  • Compare the result to the provided choices and select the correct one.

Now, let's work through each step:

Step 1: The expression given is (23)2\left(\frac{2}{3}\right)^2.

Step 2: According to the exponentiation rule, we apply the exponent to both the numerator and the denominator:
(23)2=2232\left(\frac{2}{3}\right)^2 = \frac{2^2}{3^2}.

Step 3: Calculate 222^2 and 323^2:
22=42^2 = 4, and 32=93^2 = 9.

Step 4: Form the resultant fraction:
Thus, 2232=49\frac{2^2}{3^2} = \frac{4}{9}.

Step 5: Finally, compare this result with the given choices:
Our result 49\frac{4}{9} matches with choice 2.

Therefore, the solution to the problem is 49 \frac{4}{9} .

Answer

49 \frac{4}{9}

Exercise #5

Insert the corresponding expression:

(13)3= \left(\frac{1}{3}\right)^3=

Video Solution

Step-by-Step Solution

To solve the expression (13)3 \left(\frac{1}{3}\right)^3 , we need to apply the rule for exponents of a fraction, which states:

(ab)n=anbn \left(\frac{a}{b}\right)^n = \frac{a^n}{b^n}

Using this property, we can rewrite the fraction with its exponent as follows:

(13)3=1333 \left(\frac{1}{3}\right)^3 = \frac{1^3}{3^3}

Now, calculate the powers of the numerator and the denominator separately:

  • 13=1 1^3 = 1

  • 33=27 3^3 = 27

Thus, putting it all together, we have:

1333=127 \frac{1^3}{3^3} = \frac{1}{27}

This shows that raising both the numerator and the denominator of a fraction to a power involves calculating the power of each part separately and then constructing a new fraction.

The solution to the question is: 127 \frac{1}{27}

Answer

127 \frac{1}{27}

Exercise #6

Insert the corresponding expression:

(12)2= \left(\frac{1}{2}\right)^2=

Video Solution

Step-by-Step Solution

To solve this problem, we'll follow these steps:

  • Step 1: Identify the given fraction and power: 12 \frac{1}{2} and the power 2 2 .
  • Step 2: Apply the formula (ab)n=anbn(\frac{a}{b})^n = \frac{a^n}{b^n}.
  • Step 3: Calculate the numerator as 12=1 1^2 = 1 .
  • Step 4: Calculate the denominator as 22=4 2^2 = 4 .
  • Step 5: Combine the results to find (12)2=14\left(\frac{1}{2}\right)^2 = \frac{1}{4}.

Now, let's work through each step in detail:
Step 1: The fraction given is 12 \frac{1}{2} and the power is 2.
Step 2: Applying the formula (ab)n=anbn(\frac{a}{b})^n = \frac{a^n}{b^n}, we have:
Step 3: Calculate 12=1 1^2 = 1 .
Step 4: Calculate 22=4 2^2 = 4 .
Step 5: Compile these to get 14\frac{1}{4}.

Therefore, the solution to the problem is 14 \frac{1}{4} , which matches choice 2.

Answer

14 \frac{1}{4}

Exercise #7

Insert the corresponding expression:

(45)3= \left(\frac{4}{5}\right)^3=

Video Solution

Step-by-Step Solution

To solve this problem, we will evaluate the expression (45)3\left(\frac{4}{5}\right)^3 following these steps:

  • Step 1: Apply the power to the numerator.
  • Step 2: Apply the power to the denominator.
  • Step 3: Calculate the results for both and simplify if possible.

Let's go through each step:

Step 1: Calculate the cube of the numerator, which is 4. That is, 43=4×4×4=644^3 = 4 \times 4 \times 4 = 64.

Step 2: Calculate the cube of the denominator, which is 5. That is, 53=5×5×5=1255^3 = 5 \times 5 \times 5 = 125.

Step 3: Combine these results into a fraction: 64125\frac{64}{125}.

Therefore, the expression (45)3\left(\frac{4}{5}\right)^3 simplifies to 64125\frac{64}{125}.

Upon comparing with the given choices, the correct answer is choice 1: 64125\frac{64}{125}.

Answer

64125 \frac{64}{125}

Exercise #8

Insert the corresponding expression:

(45)2= \left(\frac{4}{5}\right)^2=

Video Solution

Step-by-Step Solution

To solve the problem of finding the value of (45)2\left(\frac{4}{5}\right)^2, we will follow these steps:

  • Step 1: Identify the numerator and the denominator of the fraction. Here, the numerator is 4, and the denominator is 5.
  • Step 2: Apply the exponent to both the numerator and the denominator separately: (45)2=4252\left(\frac{4}{5}\right)^2 = \frac{4^2}{5^2}.
  • Step 3: Calculate 4252\frac{4^2}{5^2}. This gives:
    • The square of the numerator is 42=164^2 = 16.
    • The square of the denominator is 52=255^2 = 25.
    Thus, 4252=1625\frac{4^2}{5^2} = \frac{16}{25}.

Since we successfully calculated (45)2=1625\left(\frac{4}{5}\right)^2 = \frac{16}{25}, this matches the choice labeled 1625 \frac{16}{25} .

Therefore, the correct expression is 1625 \frac{16}{25} .

Answer

1625 \frac{16}{25}

Exercise #9

What is the result of the following power?

(34)2 (\frac{3}{4})^2

Step-by-Step Solution

To solve (34)2 (\frac{3}{4})^2 , you need to square both the numerator and the denominator separately:

1. Square the numerator: 32=9 3^2 = 9

2. Square the denominator: 42=16 4^2 = 16

3. Combine the results to get 916 \frac{9}{16}

Answer

916 \frac{9}{16}

Exercise #10

What is the result of the following power?

(56)2 (\frac{5}{6})^2

Step-by-Step Solution

To solve (56)2 (\frac{5}{6})^2 , you need to square both the numerator and the denominator separately:

1. Square the numerator: 52=25 5^2 = 25

2. Square the denominator: 62=36 6^2 = 36

3. Combine the results to get 2536 \frac{25}{36}

Answer

2536 \frac{25}{36}

Exercise #11

Compute the value of:

(34)3=(\frac{3}{4})^3=

Step-by-Step Solution

To compute (34)3(\frac{3}{4})^3, you must raise both the numerator and the denominator to the power of 3:

Numerator: 33=3×3×3=273^3 = 3 \times 3 \times 3 = 27

Denominator: 43=4×4×4=644^3 = 4 \times 4 \times 4 = 64

Therefore, (34)3=2764(\frac{3}{4})^3 = \frac{27}{64}.

Answer

2764\frac{27}{64}

Exercise #12

Evaluate the expression:

(56)2=(\frac{5}{6})^2=

Step-by-Step Solution

To evaluate (56)2(\frac{5}{6})^2, raise both the numerator and the denominator to the power of 2:

Numerator: 52=5×5=255^2 = 5 \times 5 = 25

Denominator: 62=6×6=366^2 = 6 \times 6 = 36

Therefore, (56)2=2536(\frac{5}{6})^2 = \frac{25}{36}.

Answer

2536\frac{25}{36}

Exercise #13

What is the result of:

(75)2=(\frac{7}{5})^2=

Step-by-Step Solution

To determine (75)2(\frac{7}{5})^2, raise both the numerator and the denominator to the power of 2:

Numerator: 72=7×7=497^2 = 7 \times 7 = 49

Denominator: 52=5×5=255^2 = 5 \times 5 = 25

Therefore, (75)2=4925(\frac{7}{5})^2 = \frac{49}{25}.

Answer

4925\frac{49}{25}

Exercise #14

What is the result of the following power?

(23)3 (\frac{2}{3})^3

Video Solution

Step-by-Step Solution

To solve the given power expression, we need to apply the formula for powers of a fraction. The expression we are given is:
(23)3 \left(\frac{2}{3}\right)^3

Let's break down the steps:

  • When we raise a fraction to a power, we apply the exponent to both the numerator and the denominator separately. This means raising both 2 and 3 to the power of 3.
  • Thus, we calculate:
    23=8 2^3 = 8 and 33=27 3^3 = 27 .
  • Therefore, (23)3=2333=827 \left(\frac{2}{3}\right)^3 = \frac{2^3}{3^3} = \frac{8}{27} .

So, the result of the expression (23)3 \left(\frac{2}{3}\right)^3 is 827 \frac{8}{27} .

Answer

827 \frac{8}{27}

Exercise #15

(26)3= (\frac{2}{6})^3=

Video Solution

Step-by-Step Solution

We use the formula:

(ab)n=anbn (\frac{a}{b})^n=\frac{a^n}{b^n}

(26)3=(22×3)3 (\frac{2}{6})^3=(\frac{2}{2\times3})^3

We simplify:

(13)3=1333 (\frac{1}{3})^3=\frac{1^3}{3^3}

1×1×13×3×3=127 \frac{1\times1\times1}{3\times3\times3}=\frac{1}{27}

Answer

127 \frac{1}{27}

Exercise #16

Insert the corresponding expression:

(a×b2×x)3= \left(\frac{a\times b}{2\times x}\right)^3=

Video Solution

Step-by-Step Solution

To solve this problem, we'll follow these steps:

  • Step 1: Express the given expression (a×b2×x)3\left(\frac{a \times b}{2 \times x}\right)^3 using exponent rules for fractions.
  • Step 2: Apply the cube power to both the numerator and the denominator separately.
  • Step 3: Simplify the expression and compare it to the provided choices.

Now, let's work through each step:

Step 1: We have the expression (a×b2×x)3\left(\frac{a \times b}{2 \times x}\right)^3. According to the power of a fraction rule, (mn)p=mpnp\left(\frac{m}{n}\right)^p = \frac{m^p}{n^p}, we can raise the numerator and denominator to the power of 3 separately.

Step 2: Apply the cube power:

  • Numerator: (a×b)3=a3×b3(a \times b)^3 = a^3 \times b^3
  • Denominator: (2×x)3=23×x3=8×x3(2 \times x)^3 = 2^3 \times x^3 = 8 \times x^3

Step 3: Combine these results:

The expression simplifies to a3×b38×x3\frac{a^3 \times b^3}{8 \times x^3}.

Comparing it to the given choices, we find that this matches Choice 1: a3×b38×x3\frac{a^3 \times b^3}{8 \times x^3}.

Therefore, the solution to the problem is a3×b38×x3\frac{a^3 \times b^3}{8 \times x^3}.

Answer

a3×b38×x3 \frac{a^3\times b^3}{8\times x^3}

Exercise #17

Insert the corresponding expression:

(a×32×x)3= \left(\frac{a\times3}{2\times x}\right)^3=

Video Solution

Step-by-Step Solution

To solve this problem, we'll follow these steps:

  • Step 1: Identify the given expression and its structure.
  • Step 2: Apply the rule for powers of a fraction, (pq)n\left(\frac{p}{q}\right)^n.
  • Step 3: Calculate powers of individual components within the fraction.
  • Step 4: Simplify the resulting expression.

Now, let's work through each step:

Step 1: Our given expression is (a×32×x)3\left(\frac{a \times 3}{2 \times x}\right)^3.

Step 2: Apply the power to the entire fraction using (pq)n=pnqn\left(\frac{p}{q}\right)^n = \frac{p^n}{q^n}, we get:
(a×32×x)3=(a×3)3(2×x)3 \left(\frac{a \times 3}{2 \times x}\right)^3 = \frac{(a \times 3)^3}{(2 \times x)^3} .

Step 3: Simplify the numerator and the denominator separately:
Numerator: (a×3)3=a3×33=a3×27(a \times 3)^3 = a^3 \times 3^3 = a^3 \times 27.
Denominator: (2×x)3=23×x3=8×x3(2 \times x)^3 = 2^3 \times x^3 = 8 \times x^3.

Step 4: Combine the simplified components to form the final expression:
The expression is a3×278×x3\frac{a^3 \times 27}{8 \times x^3}.

Therefore, the solution to the problem is a3×278×x3 \frac{a^3 \times 27}{8 \times x^3} , which corresponds to choice 2.

Answer

a3×278×x3 \frac{a^3\times27}{8\times x^3}

Exercise #18

Insert the corresponding expression:

(6x×y)2= \left(\frac{6}{x\times y}\right)^2=

Video Solution

Step-by-Step Solution

To solve this problem, we'll apply the rule for powers of a fraction:

  • Step 1: Identify the fraction. The fraction given is 6x×y\frac{6}{x \times y}.
  • Step 2: Apply the power to both the numerator and the denominator. This means squaring both 6 and x×yx \times y.
  • Step 3: Calculate the square of the numerator and the denominator:
    • The square of the numerator: 62=366^2 = 36.
    • The square of the denominator: (x×y)2=x2×y2(x \times y)^2 = x^2 \times y^2.
  • Step 4: Combine the results: (6x×y)2=62(x×y)2=36x2×y2\left(\frac{6}{x \times y}\right)^2 = \frac{6^2}{(x \times y)^2} = \frac{36}{x^2 \times y^2}.

Thus, the expression (6x×y)2\left(\frac{6}{x \times y}\right)^2 simplifies to 36(x×y)2\frac{36}{(x \times y)^2}.

Therefore, the correct answer is clearly the expression 36(x×y)2\frac{36}{(x \times y)^2}, which matches choice 4.

Answer

36(x×y)2 \frac{36}{\left(x\times y\right)^2}

Exercise #19

Insert the corresponding expression:

(2×a3)2= \left(\frac{2\times a}{3}\right)^2=

Video Solution

Step-by-Step Solution

The task is to simplify (2×a3)2\left(\frac{2\times a}{3}\right)^2.

First, applying the exponent rule for fractions, (bc)n=bncn\left(\frac{b}{c}\right)^n = \frac{b^n}{c^n}, we have:

  • (2×a3)2=(2×a)232\left(\frac{2\times a}{3}\right)^2 = \frac{(2 \times a)^2}{3^2} - which represents performing the exponentiation separately on the numerator and denominator.

Now, simplify each part:

  • The numerator: (2×a)2=22×a2=4×a2(2 \times a)^2 = 2^2 \times a^2 = 4 \times a^2.
  • The denominator: 32=93^2 = 9.

Thus, the expression simplifies to 4×a29\frac{4 \times a^2}{9}.

We ensure the valid transformations based on the choices provided:

  • Choice 1: 4×a29\frac{4\times a^2}{9}, which matches what we calculated.
  • Choice 2: 22×a232\frac{2^2\times a^2}{3^2}, which is an equivalent form before final multiplication.
  • Choice 3: (2×a)232\frac{\left(2\times a\right)^2}{3^2}, presenting the step before breaking down the (2a)2 (2a)^2 .
  • Choice 4: "All answers are correct", recognizing all transformations as valid.

Therefore, each choice represents correct steps or forms towards the simplified expression.

The correct answer is: All answers are correct.

Answer

All answers are correct

Exercise #20

3004(1300)4=? 300^{-4}\cdot(\frac{1}{300})^{-4}=?

Video Solution

Step-by-Step Solution

To solve the problem 3004(1300)4 300^{-4} \cdot \left(\frac{1}{300}\right)^{-4} , let's follow these steps:

  • Step 1: Simplify the reciprocal power expression.
    The expression (1300)4 \left(\frac{1}{300}\right)^{-4} can be simplified using the rule for reciprocals, which states (1a)n=an \left(\frac{1}{a}\right)^{-n} = a^n . Thus, (1300)4=3004\left(\frac{1}{300}\right)^{-4} = 300^4.
  • Step 2: Combine the powers.
    Now the expression becomes 30043004 300^{-4} \cdot 300^4 . Using the rule aman=am+n a^m \cdot a^n = a^{m+n} , we have 3004+4=3000 300^{-4+4} = 300^0 .
  • Step 3: Calculate the final result.
    By the identity a0=1 a^0 = 1 , any non-zero number raised to the power of zero equals 1. Therefore, 3000=1 300^0 = 1 .

Thus, the solution to the problem is 1 \boxed{1} .

Answer

1