Powers of a Fraction: Calculating powers with negative exponents

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

Insert the corresponding expression:

\( \)\( \left(\frac{7\times11\times19}{3\times12\times15}\right)^{-4}= \)

Question 2

\( (\frac{2}{3})^{-4}=\text{?} \)

Question 3

Insert the corresponding expression:

\( \left(\frac{4}{9}\right)^{-5}= \)

Question 4

Insert the corresponding expression:

\( \left(\frac{10}{13}\right)^{-4}= \)

Question 5

Insert the corresponding expression:

\( \left(\frac{2}{3}\right)^{-11}= \)

Examples with solutions for Powers of a Fraction: Calculating powers with negative exponents

Exercise #1

Insert the corresponding expression:

(7×11×193×12×15)−4= \left(\frac{7\times11\times19}{3\times12\times15}\right)^{-4}=

Video Solution

Step-by-Step Solution

The given expression is:
(7×11×193×12×15)−4 \left(\frac{7\times11\times19}{3\times12\times15}\right)^{-4}

To solve this expression, we need to apply the rules of exponents, specifically the rule for powers of a fraction. For any fraction(ab)−n \left(\frac{a}{b}\right)^{-n} , the expression is equivalent to(ba)n \left(\frac{b}{a}\right)^n .
Therefore, negative exponents indicate that the fraction should be flipped and raised to the positive of that exponent.

Substitute the terms into this formula:
1. Flip the fraction: (3×12×157×11×19) \left(\frac{3\times12\times15}{7\times11\times19}\right)
2. Raise both numerator and denominator to the power of 4:
Thus, we have:
(3×12×157×11×19)4 \left(\frac{3\times12\times15}{7\times11\times19}\right)^{4}

Now evaluating each term individually:
- In the numerator:
- 34×124×154 3^4\times12^4\times15^4
- In the denominator:
- 74×114×194 7^4\times11^4\times19^4

Applying the negative exponent rule, each individual factor in both numerator and denominator should be inverted, altering the exponents to negative:
1. Numerator becomes: 3−4×12−4×15−4 3^{-4}\times12^{-4}\times15^{-4}
2. Denominator becomes: 7−4×11−4×19−4 7^{-4}\times11^{-4}\times19^{-4}

Rewriting the expression, we achieve:
7−4×11−4×19−43−4×12−4×15−4 \frac{7^{-4}\times11^{-4}\times19^{-4}}{3^{-4}\times12^{-4}\times15^{-4}}

This matches precisely the provided solution.

The solution to the question is:7−4×11−4×19−43−4×12−4×15−4 \frac{7^{-4}\times11^{-4}\times19^{-4}}{3^{-4}\times12^{-4}\times15^{-4}}

Answer

7−4×11−4×19−43−4×12−4×15−4 \frac{7^{-4}\times11^{-4}\times19^{-4}}{3^{-4}\times12^{-4}\times15^{-4}}

Exercise #2

(23)−4=? (\frac{2}{3})^{-4}=\text{?}

Video Solution

Step-by-Step Solution

We use the formula:

(ab)−n=(ba)n (\frac{a}{b})^{-n}=(\frac{b}{a})^n

Therefore, we obtain:

(32)4 (\frac{3}{2})^4

We use the formula:

(ba)n=bnan (\frac{b}{a})^n=\frac{b^n}{a^n}

Therefore, we obtain:

3424=3×3×3×32×2×2×2=8116 \frac{3^4}{2^4}=\frac{3\times3\times3\times3}{2\times2\times2\times2}=\frac{81}{16}

Answer

8116 \frac{81}{16}

Exercise #3

Insert the corresponding expression:

(49)−5= \left(\frac{4}{9}\right)^{-5}=

Video Solution

Answer

4−59−5 \frac{4^{-5}}{9^{-5}}

Exercise #4

Insert the corresponding expression:

(1013)−4= \left(\frac{10}{13}\right)^{-4}=

Video Solution

Answer

10−413−4 \frac{10^{-4}}{13^{-4}}

Exercise #5

Insert the corresponding expression:

(23)−11= \left(\frac{2}{3}\right)^{-11}=

Video Solution

Answer

2−113−11 \frac{2^{-11}}{3^{-11}}

Question 1

Insert the corresponding expression:

\( \left(\frac{1}{4}\right)^{-2}= \)

Question 2

Insert the corresponding expression:

\( \left(\frac{2}{5}\right)^{-3}= \)

Question 3

Insert the corresponding expression:

\( \left(\frac{3}{7}\right)^{-4}= \)

Question 4

Insert the corresponding expression:

\( \left(\frac{1}{20}\right)^{-7}= \)

Question 5

Insert the corresponding expression:

\( \left(\frac{5}{7}\right)^{-7}= \)

Exercise #6

Insert the corresponding expression:

(14)−2= \left(\frac{1}{4}\right)^{-2}=

Video Solution

Answer

1−24−2 \frac{1^{-2}}{4^{-2}}

Exercise #7

Insert the corresponding expression:

(25)−3= \left(\frac{2}{5}\right)^{-3}=

Video Solution

Answer

2−35−3 \frac{2^{-3}}{5^{-3}}

Exercise #8

Insert the corresponding expression:

(37)−4= \left(\frac{3}{7}\right)^{-4}=

Video Solution

Answer

3−47−4 \frac{3^{-4}}{7^{-4}}

Exercise #9

Insert the corresponding expression:

(120)−7= \left(\frac{1}{20}\right)^{-7}=

Video Solution

Answer

207 20^7

Exercise #10

Insert the corresponding expression:

(57)−7= \left(\frac{5}{7}\right)^{-7}=

Video Solution

Answer

(75)7 \left(\frac{7}{5}\right)^7

Question 1

Insert the corresponding expression:

\( \left(\frac{10}{13}\right)^{-2}= \)

Question 2

Insert the corresponding expression:

\( \left(\frac{15}{21}\right)^{-3}= \)

Question 3

Insert the corresponding expression:

\( \left(\frac{1}{60}\right)^{-4}= \)

Question 4

Insert the corresponding expression:

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

Question 5

Insert the corresponding expression:

\( \)\( \left(\frac{1}{3}\right)^{-4}= \)

Exercise #11

Insert the corresponding expression:

(1013)−2= \left(\frac{10}{13}\right)^{-2}=

Video Solution

Answer

(1310)2 \left(\frac{13}{10}\right)^2

Exercise #12

Insert the corresponding expression:

(1521)−3= \left(\frac{15}{21}\right)^{-3}=

Video Solution

Answer

(2115)3 \left(\frac{21}{15}\right)^3

Exercise #13

Insert the corresponding expression:

(160)−4= \left(\frac{1}{60}\right)^{-4}=

Video Solution

Answer

604 60^4

Exercise #14

Insert the corresponding expression:

(25)−2= \left(\frac{2}{5}\right)^{-2}=

Video Solution

Answer

(52)2 \left(\frac{5}{2}\right)^2

Exercise #15

Insert the corresponding expression:

(13)−4= \left(\frac{1}{3}\right)^{-4}=

Video Solution

Answer

34 3^4

Question 1

Insert the corresponding expression:

\( \left(\frac{3}{8}\right)^{-5}= \)

Question 2

Insert the corresponding expression:

\( \left(\frac{5}{6}\right)^{-3}= \)

Question 3

Insert the corresponding expression:

\( \left(\frac{10}{17}\right)^{-5}= \)

Question 4

Insert the corresponding expression:

\( \left(\frac{8}{17\times13}\right)^{-6}= \)

Question 5

Insert the corresponding expression:

\( \left(\frac{1\times5}{4\times6}\right)^{-2}= \)

Exercise #16

Insert the corresponding expression:

(38)−5= \left(\frac{3}{8}\right)^{-5}=

Video Solution

Answer

(83)5 \left(\frac{8}{3}\right)^5

Exercise #17

Insert the corresponding expression:

(56)−3= \left(\frac{5}{6}\right)^{-3}=

Video Solution

Answer

(65)3 \left(\frac{6}{5}\right)^3

Exercise #18

Insert the corresponding expression:

(1017)−5= \left(\frac{10}{17}\right)^{-5}=

Video Solution

Answer

(1710)5 \left(\frac{17}{10}\right)^5

Exercise #19

Insert the corresponding expression:

(817×13)−6= \left(\frac{8}{17\times13}\right)^{-6}=

Video Solution

Answer

A+B are correct

Exercise #20

Insert the corresponding expression:

(1×54×6)−2= \left(\frac{1\times5}{4\times6}\right)^{-2}=

Video Solution

Answer

a' + b' are correct

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