Powers of a Fraction: Variable in the base of the power

Combination of different basesComplete the equationPresenting powers in the denominator as powers with negative exponentsUsing multiple rulesApplying the formulaVariable in the exponent 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:

\( \frac{a^5\times x^5}{7^5\times b^5}= \)

Question 2

Insert the corresponding expression:

\( \frac{1}{a^{-x}}= \)

Question 3

Insert the corresponding expression:

\( \left(\frac{x}{y}\right)^8= \)

Question 4

Insert the corresponding expression:

\( \left(\frac{a}{b}\right)^9= \)

Question 5

Insert the corresponding expression:

\( \left(\frac{4}{a\times b}\right)^2= \)

Examples with solutions for Powers of a Fraction: Variable in the base of the power

Exercise #1

Insert the corresponding expression:

a5×x575×b5= \frac{a^5\times x^5}{7^5\times b^5}=

Video Solution

Step-by-Step Solution

To solve this problem, our goal is to express the given quotient of powers in a simplified form using exponent laws.

  • Step 1: Understand the original expression. We have a5×x575×b5\frac{a^5 \times x^5}{7^5 \times b^5}.
  • Step 2: Recognize the structure. Notice that both the numerator and denominator are raised to the fifth power.
  • Step 3: Apply the property of exponents for quotients and products, which states that (mn)k=mknk\left(\frac{m}{n}\right)^k = \frac{m^k}{n^k} and (m⋅n)k=mk⋅nk(m \cdot n)^k = m^k \cdot n^k.
  • Step 4: Rewrite the expression as a single fraction raised to the power of 5. Since each term in the numerator and denominator is raised to the fifth power separately, we combine them under a single power:
    • Numerator: a5×x5=(a×x)5a^5 \times x^5 = (a \times x)^5
    • Denominator: 75×b5=(7×b)57^5 \times b^5 = (7 \times b)^5
    • Therefore, a5×x575×b5=(a×x7×b)5\frac{a^5 \times x^5}{7^5 \times b^5} = \left(\frac{a \times x}{7 \times b}\right)^5.

Thus, the expression can be written as: (a×x7×b)5\left(\frac{a \times x}{7 \times b}\right)^5.

Now, comparing this with the answer choices provided:

  • Choice 1: (a×x)57×b5\frac{(a \times x)^5}{7 \times b^5} - does not match, as it retains the separate powers incorrectly.
  • Choice 2: (a×x7×b)5\left(\frac{a \times x}{7 \times b}\right)^5 - matches perfectly as derived.
  • Choice 3: a×x5(7×b)5\frac{a \times x^5}{(7 \times b)^5} - incorrect form compared to derived structure.
  • Choice 4: 7×(a×xb)57 \times \left(\frac{a \times x}{b}\right)^5 - unrelated format, doesn't match.

The correct choice is therefore Choice 2. This matches our derived expression using the laws of exponents correctly.

Answer

(a×x7×b)5 \left(\frac{a\times x}{7\times b}\right)^5

Exercise #2

Insert the corresponding expression:

1a−x= \frac{1}{a^{-x}}=

Video Solution

Step-by-Step Solution

We begin with the expression: 1a−x \frac{1}{a^{-x}} .
Our goal is to simplify this expression while converting any negative exponents into positive ones.

  • Recall the rule for negative exponents: a−n=1an a^{-n} = \frac{1}{a^n} .
  • Correspondingly, 1a−n=an \frac{1}{a^{-n}} = a^n .
  • Thus, in our expression 1a−x \frac{1}{a^{-x}} , the negative exponent can be converted and flipped to the numerator by the rule: 1a−x=ax \frac{1}{a^{-x}} = a^x .
Therefore, the expression evaluates to ax a^x .

The solution to the question is: ax a^x .

Answer

ax a^x

Exercise #3

Insert the corresponding expression:

(xy)8= \left(\frac{x}{y}\right)^8=

Video Solution

Answer

x8y8 \frac{x^8}{y^8}

Exercise #4

Insert the corresponding expression:

(ab)9= \left(\frac{a}{b}\right)^9=

Video Solution

Answer

a9b9 \frac{a^9}{b^9}

Exercise #5

Insert the corresponding expression:

(4a×b)2= \left(\frac{4}{a\times b}\right)^2=

Video Solution

Answer

42(a×b)2 \frac{4^2}{\left(a\times b\right)^2}

Question 1

Insert the corresponding expression:

\( \left(\frac{5}{x\times y}\right)^5= \)

Question 2

Insert the corresponding expression:

\( \left(\frac{a\times x}{7}\right)^6= \)

Question 3

Insert the corresponding expression:

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

Question 4

Insert the corresponding expression:

\( \left(\frac{x\times a}{b\times y}\right)^4= \)

Question 5

Insert the corresponding expression:

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

Exercise #6

Insert the corresponding expression:

(5x×y)5= \left(\frac{5}{x\times y}\right)^5=

Video Solution

Answer

a'+b' are correct

Exercise #7

Insert the corresponding expression:

(a×x7)6= \left(\frac{a\times x}{7}\right)^6=

Video Solution

Answer

a6×x676 \frac{a^6\times x^6}{7^6}

Exercise #8

Insert the corresponding expression:

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

Video Solution

Answer

A'+C' are correct

Exercise #9

Insert the corresponding expression:

(x×ab×y)4= \left(\frac{x\times a}{b\times y}\right)^4=

Video Solution

Answer

x4×a4(b×y)4 \frac{x^4\times a^4}{\left(b\times y\right)^4}

Exercise #10

Insert the corresponding expression:

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

Video Solution

Answer

a232 \frac{a^2}{3^2}

Question 1

Insert the corresponding expression:

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

Question 2

Insert the corresponding expression:

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

Question 3

\( \)

Insert the corresponding expression:

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

Question 4

Insert the corresponding expression:

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

Question 5

Insert the corresponding expression:

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

Exercise #11

Insert the corresponding expression:

(b5)4= \left(\frac{b}{5}\right)^4=

Video Solution

Answer

b454 \frac{b^4}{5^4}

Exercise #12

Insert the corresponding expression:

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

Video Solution

Answer

63x3 \frac{6^3}{x^3}

Exercise #13

Insert the corresponding expression:

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

Video Solution

Answer

57y7 \frac{5^7}{y^7}

Exercise #14

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 #15

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}

Question 1

Insert the corresponding expression:

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

Question 2

Insert the corresponding expression:

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

Question 3

Insert the corresponding expression:

\( \frac{a^7\times x^7}{5^7}= \)

Question 4

Insert the corresponding expression:

\( \frac{3^8\times a^8}{x^8\times5^8}= \)

Question 5

Insert the corresponding expression:

\( \frac{4^9\times b^9}{x^9\times y^9}= \)

Exercise #16

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 #17

Insert the corresponding expression:

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

Video Solution

Answer

All answers are correct

Exercise #18

Insert the corresponding expression:

a7×x757= \frac{a^7\times x^7}{5^7}=

Video Solution

Answer

A+B are correct

Exercise #19

Insert the corresponding expression:

38×a8x8×58= \frac{3^8\times a^8}{x^8\times5^8}=

Video Solution

Answer

(3×ax×5)8 \left(\frac{3\times a}{x\times5}\right)^8

Exercise #20

Insert the corresponding expression:

49×b9x9×y9= \frac{4^9\times b^9}{x^9\times y^9}=

Video Solution

Answer

a'+b' are correct