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 formulaNumber of termsconverting Negative Exponents to Positive ExponentsCalculating powers with negative exponentsSystem of equations with no solution
Question 1

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

Question 2

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

Question 3

\( 5^4\cdot(\frac{1}{5})^4=\text{?} \)

Question 4

Insert the corresponding expression:

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

Question 5

Insert the corresponding expression:

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

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

Exercise #1

(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 #2

(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 #3

54⋅(15)4=? 5^4\cdot(\frac{1}{5})^4=\text{?}

Video Solution

Step-by-Step Solution

This problem can be solved using the Law of exponents power rules for a negative power, power over a power, as well as the power rule for the product between terms with identical bases.

However we prefer to solve it in a quicker way:

To this end, the power by power law is applied to the parentheses in which the terms are multiplied, but in the opposite direction:

xn⋅yn=(x⋅y)n x^n\cdot y^n=(x\cdot y)^n Since in the expression in the problem there is a multiplication between two terms with identical powers, this law can be used in its opposite sense.

54⋅(15)4=(5⋅15)4 5^4\cdot(\frac{1}{5})^4=\big(5\cdot\frac{1}{5}\big)^4 Since the multiplication in the given problem is between terms with the same power, we can apply this law in the opposite direction and write the expression as the multiplication of the bases of the terms in parentheses to which the same power is applied.

We continue and simplify the expression inside of the parentheses. We can do it quickly if inside the parentheses there is a multiplication between two opposite numbers, then their product will give the result: 1, All of the above is applied to the problem leading us to the last step:

(5⋅15)4=14=1 \big(5\cdot\frac{1}{5}\big)^4 = 1^4=1 We remember that raising the number 1 to any power will always give the result: 1, which means that:

1x=1 1^x=1 Summarizing the steps to solve the problem, we obtain the following:

54⋅(15)4=(5⋅15)4=1 5^4\cdot(\frac{1}{5})^4=\big(5\cdot\frac{1}{5}\big)^4 =1 Therefore, the correct answer is option b.

Answer

1

Exercise #4

Insert the corresponding expression:

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

Video Solution

Answer

3664 \frac{36}{64}

Exercise #5

Insert the corresponding expression:

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

Video Solution

Answer

278 \frac{27}{8}

Question 1

Insert the corresponding expression:

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

Question 2

Insert the corresponding expression:

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

Question 3

Insert the corresponding expression:

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

Question 4

Insert the corresponding expression:

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

Question 5

Insert the corresponding expression:

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

Exercise #6

Insert the corresponding expression:

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

Video Solution

Answer

1649 \frac{16}{49}

Exercise #7

Insert the corresponding expression:

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

Video Solution

Answer

49 \frac{4}{9}

Exercise #8

Insert the corresponding expression:

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

Video Solution

Answer

14 \frac{1}{4}

Exercise #9

Insert the corresponding expression:

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

Video Solution

Answer

64125 \frac{64}{125}

Exercise #10

Insert the corresponding expression:

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

Video Solution

Answer

1625 \frac{16}{25}

Question 1

What is the result of the following power?

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

Question 2

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

Question 3

Insert the corresponding expression:

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

Question 4

Insert the corresponding expression:

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

Question 5

Insert the corresponding expression:

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

Exercise #11

What is the result of the following power?

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

Video Solution

Answer

827 \frac{8}{27}

Exercise #12

(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 #13

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 #14

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 #15

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}

Question 1

Insert the corresponding expression:

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

Question 2

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

Exercise #16

Insert the corresponding expression:

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

Video Solution

Answer

All answers are correct

Exercise #17

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

Video Solution

Answer

1