Powers of a Fraction: Applying the formula

Combination of different basesComplete the equationPresenting powers in the denominator as powers with negative exponentsUsing multiple rulesVariable in the exponent of the powerVariable in the base of the powerInverse formulaconverting Negative Exponents to Positive ExponentsNumber of termsCalculating powers with negative exponentsSystem of equations with no solution
Question 1

Insert the corresponding expression:

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

Question 2

What is the result of the following power?

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

Question 3

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

Question 4

\( (\frac{4^2}{7^4})^2= \)

Question 5

Insert the corresponding expression:

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

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

Exercise #1

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 #2

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 #3

(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 #4

(4274)2= (\frac{4^2}{7^4})^2=

Video Solution

Step-by-Step Solution

(4274)2=42×274×2=4478 (\frac{4^2}{7^4})^2=\frac{4^{2\times2}}{7^{4\times2}}=\frac{4^4}{7^8}

Answer

4478 \frac{4^4}{7^8}

Exercise #5

Insert the corresponding expression:

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

Video Solution

Answer

3664 \frac{36}{64}

Question 1

Insert the corresponding expression:

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

Question 2

Insert the corresponding expression:

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

Question 3

Insert the corresponding expression:

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

Question 4

Insert the corresponding expression:

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

Question 5

Insert the corresponding expression:

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

Exercise #6

Insert the corresponding expression:

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

Video Solution

Answer

278 \frac{27}{8}

Exercise #7

Insert the corresponding expression:

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

Video Solution

Answer

1649 \frac{16}{49}

Exercise #8

Insert the corresponding expression:

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

Video Solution

Answer

49 \frac{4}{9}

Exercise #9

Insert the corresponding expression:

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

Video Solution

Answer

14 \frac{1}{4}

Exercise #10

Insert the corresponding expression:

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

Video Solution

Answer

64125 \frac{64}{125}

Question 1

Insert the corresponding expression:

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

Question 2

Insert the corresponding expression:

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

Question 3

Insert the corresponding expression:

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

Question 4

Insert the corresponding expression:

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

Question 5

Insert the corresponding expression:

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

Exercise #11

Insert the corresponding expression:

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

Video Solution

Answer

1625 \frac{16}{25}

Exercise #12

Insert the corresponding expression:

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

Video Solution

Answer

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

Exercise #13

Insert the corresponding expression:

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

Video Solution

Answer

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

Exercise #14

Insert the corresponding expression:

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

Video Solution

Answer

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

Exercise #15

Insert the corresponding expression:

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

Video Solution

Answer

All answers are correct

Question 1

\( (\frac{13}{2})^0\cdot(\frac{2}{13})^{-2}\cdot(\frac{13}{2})^{-5}=\text{?} \)

Question 2

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

Exercise #16

(132)0⋅(213)−2⋅(132)−5=? (\frac{13}{2})^0\cdot(\frac{2}{13})^{-2}\cdot(\frac{13}{2})^{-5}=\text{?}

Video Solution

Answer

(213)3 (\frac{2}{13})^3

Exercise #17

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

Video Solution

Answer

1

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