Insert the corresponding expression: $$ \left(\frac{7\times13}{6\times10\times12}\right)^{-3}= $$

$$ \frac{12^{-3}\times10^{-3}\times6^{-3}}{7^{-3}\times13^{-3}} $$

$$ \frac{7^{-3}\times13^{-3}}{6^{-3}\times10^{-3}\times12^{-3}} $$

$$ \frac{\left(7\times13\right)^{-3}}{\left(6\times10\times12\right)^{-3}} $$

Insert the corresponding expression: $$ \left(\frac{4\times8}{5\times17\times3}\right)^{-4}= $$

$$ \frac{\left(4\times8\right)^{-4}}{\left(5\times17\times3\right)^{-4}} $$

$$ \frac{\left(4\times8\right)^{-4}}{\left(5\times17\times3\right)^4} $$

$$ \frac{\left(4\times8\right)^4}{\left(5\times17\times3\right)^{-4}} $$

Insert the corresponding expression: $$ $$$$ \left(\frac{5\times3}{8\times4\times2}\right)^{-5}= $$

$$ \frac{5^{-5}\times3^{-5}}{\left(8\times4\times2\right)^{-5}} $$

$$ \frac{5^{-5}\times3^{-5}}{8^{-5}\times4^{-5}\times2^{-5}} $$

None of the answers are correct

Insert the corresponding expression: $$ \left(\frac{3\times5\times7}{2\times4\times6}\right)^{-2}= $$

$$ \frac{3^{-2}\times5^{-2}\times7^{-2}}{2^{-2}\times4^{-2}\times6^{-2}} $$

$$ \frac{\left(3\times5\times7\right)^{-2}}{\left(2\times4\times6\right)^2} $$

$$ \frac{3^{-2}\times5^{-2}\times7^{-2}}{2^{-2}\times4^{-2}\times6^{-2}} $$

$$ \frac{\left(3\times5\times7\right)^2}{\left(2\times4\times6\right)^{-2}} $$

$$ \frac{3\times5\times7^{-2}}{2\times4\times6^{-2}} $$

Insert the corresponding expression: $$ \left(\frac{1}{5\times6\times7}\right)^{-3}= $$

$$ \frac{1^{-3}}{5^{-3}\times6^{-3}\times7^{-3}} $$

$$ \frac{1}{5^3\times6^3\times7^3} $$

$$ -\frac{1}{5^{-3}\times6^{-3}\times7^{-3}} $$

$$ \frac{1^{-3}}{5^{-3}\times6^{-3}\times7^{-3}} $$

$$ \frac{1^{-3}}{5^3\times6^3\times7^3} $$

Insert the corresponding expression: $$ \left(\frac{1}{4\times6\times9}\right)^{-4}= $$

$$ \frac{1^{-4}}{4^{-4}\times6^{-4}\times9^{-4}} $$

$$ \frac{1^{-4}}{4^4\times6^4\times9^4} $$

$$ -\frac{1}{4^{-4}\times6^{-4}\times9^{-4}} $$

$$ -\left(4\times6\times9\right)^4 $$

$$ \frac{1^{-4}}{4^{-4}\times6^{-4}\times9^{-4}} $$

Insert the corresponding expression: $$ \left(\frac{3}{5\times8\times7}\right)^{-2}= $$

$$ \frac{3^{-2}}{5^{-2}\times8^{-2}\times7^{-2}} $$

$$ \frac{3^{-2}}{5^2\times8^2\times7^2} $$

$$ \frac{3^{-2}}{5^{-2}\times8^{-2}\times7^{-2}} $$

$$ \frac{3^2}{5^{-2}\times8^{-2}\times7^{-2}} $$

$$ -\frac{3^2}{5^2\times8^2\times7^2} $$

Insert the corresponding expression: $$ \left(\frac{5}{6\times9\times11}\right)^{-3}= $$

$$ \frac{5^{-3}}{6^{-3}\times9^{-3}\times11^{-3}} $$

$$ \frac{5^3}{6^{-3}\times9^{-3}\times11^{-3}} $$

$$ \frac{5^{-3}}{6^3\times9^3\times11^3} $$

Insert the corresponding expression: $$ \left(\frac{5\times11}{4\times7\times9}\right)^{-2}= $$

$$ \frac{5^{-2}\times11^{-2}}{4^{-2}\times7^{-2}\times9^{-2}} $$

$$ \frac{5^{-2}\times11}{4^{-2}\times7\times9} $$

$$ \frac{5\times11^{-2}}{4\times7\times9^{-2}} $$

$$ \frac{5^{-2}\times11^{-2}}{4^{-2}\times7^{-2}\times9^{-2}} $$

$$ \frac{\left(5\times11\right)^{-2}}{\left(4\times7\times9\right)^2} $$

Insert the corresponding expression: $$ \left(\frac{5\times8\times6}{21\times23\times19}\right)^{-2}= $$

$$ \frac{\left(5\times8\times6\right)^{-2}}{\left(21\times23\times19\right)^{-2}} $$

$$ \frac{5\times8\times6^{-2}}{21\times23\times19^{-2}} $$

$$ \frac{\left(5\times8\times6\right)^{-2}}{\left(21\times23\times19\right)^{-2}} $$

$$ \frac{5^{-2}\times8^{-2}\times6^{-2}}{21^2\times23^2\times19^2} $$

Insert the corresponding expression: $$ \left(\frac{4\times7\times8}{5\times9\times11}\right)^{-3}= $$

$$ \frac{\left(4\times7\times8\right)^{-3}}{\left(5\times9\times11\right)^{-3}} $$

$$ \frac{4^3\times7^3\times8^3}{5^{-3}\times9^{-3}\times11^{-3}} $$

$$ \frac{4^{-3}\times7^{-3}\times8^{-3}}{5^{-3}\times9^{-3}\times11^{-3}} $$

Insert the corresponding expression: $$ \left(\frac{10\times3}{7\times9}\right)^{-4}= $$

$$ \frac{7^4\times9^4}{10^4\times3^4} $$

$$ \frac{7^4\times9^4}{10^{-4}\times3^{-4}} $$

$$ \frac{7^4\times9^4}{\left(10\times3\right)^{-4}} $$

$$ \frac{7^4\times9^4}{10^4\times3^4} $$

Insert the corresponding expression: $$ $$$$ \left(\frac{6\times8}{2\times7}\right)^{-5}= $$

$$ \left(\frac{2\times7}{6\times8}\right)^5 $$

$$ -\left(\frac{6\times8}{2\times7}\right)^5 $$

$$ -\left(\frac{2\times7}{6\times8}\right)^{-5} $$

None of the answers are correct

Insert the corresponding expression: $$ \left(\frac{3\times7}{4\times6}\right)^{-6}= $$

$$ \frac{\left(4\times6\right)^6}{\left(3\times7\right)^6} $$

$$ $$$$ -\left(\frac{4\times6}{3\times7}\right)^6 $$

$$ \frac{3\times7^6}{4\times6^6} $$

$$ \frac{\left(4\times6\right)^6}{\left(3\times7\right)^6} $$

Powers of a Fraction: Number of terms

Exercise #1

Insert the corresponding expression:

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

Video Solution

Step-by-Step Solution

The given expression is:
$\left(\frac{7\times11\times19}{3\times12\times15}\right)^{-4}$

To solve this expression, we need to apply the rules of exponents, specifically the rule for powers of a fraction. For any fraction $\left(\frac{a}{b}\right)^{-n}$ , the expression is equivalent to $\left(\frac{b}{a}\right)^n$ .
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: $\left(\frac{3\times12\times15}{7\times11\times19}\right)$
2. Raise both numerator and denominator to the power of 4:
Thus, we have:
$\left(\frac{3\times12\times15}{7\times11\times19}\right)^{4}$

Now evaluating each term individually:
- In the numerator:
- $3^4\times12^4\times15^4$
- In the denominator:
- $7^4\times11^4\times19^4$

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

Rewriting the expression, we achieve:
$\frac{7^{-4}\times11^{-4}\times19^{-4}}{3^{-4}\times12^{-4}\times15^{-4}}$

This matches precisely the provided solution.

The solution to the question is: $\frac{7^{-4}\times11^{-4}\times19^{-4}}{3^{-4}\times12^{-4}\times15^{-4}}$

Answer

$\frac{7^{-4}\times11^{-4}\times19^{-4}}{3^{-4}\times12^{-4}\times15^{-4}}$

Exercise #2

Insert the corresponding expression:

$\left(\frac{1}{2\times3\times4}\right)^{-2}=$

Video Solution

Step-by-Step Solution

We are given the expression: $\left(\frac{1}{2\times3\times4}\right)^{-2}$ . We need to simplify it using the rules of exponents.

Step 1: Identify the base of the exponent.
The base is $\frac{1}{2\times3\times4}$ .
Step 2: Apply the rule for negative exponents.
For a fraction $\frac{1}{a}$ with a negative exponent, $\left( \frac{1}{a} \right)^{-n} = a^n$ . Therefore, $\left(\frac{1}{2\times3\times4}\right)^{-2} = (2\times3\times4)^2$
Step 3: Expand the expression.
$(2\times3\times4)^2 = 2^2 \times 3^2 \times 4^2$ .

Thus, the simplified expression is: $2^2\times3^2\times4^2$

Answer

$2^2\times3^2\times4^2$

Exercise #3

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

Video Solution

Answer

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

Exercise #4

Insert the corresponding expression:

$\left(\frac{3\times5\times7}{4\times8\times10}\right)^5=$

Video Solution

Answer

a'+b' are correct

Exercise #5

Insert the corresponding expression:

$\left(\frac{11\times5\times4}{9\times13\times17}\right)^8=$

Video Solution

Answer

$\frac{11^8\times5^8\times4^8}{9^8\times13^8\times17^8}$

Powers of a Fraction: Number of terms

Examples with solutions for Powers of a Fraction: Number of terms

Exercise #1

Video Solution

Step-by-Step Solution

Answer

Exercise #2

Video Solution

Step-by-Step Solution

Answer

Exercise #3

Video Solution

Answer

Exercise #4

Video Solution

Answer

Exercise #5

Video Solution

Answer

Exercise #6

Video Solution

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Exercise #7

Video Solution

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Exercise #8

Video Solution

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Exercise #9

Video Solution

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Exercise #10

Video Solution

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Exercise #11

Video Solution

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Exercise #12

Video Solution

Answer

Exercise #13

Video Solution

Answer

Exercise #14

Video Solution

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Exercise #15

Video Solution

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Exercise #16

Video Solution

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Exercise #17

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Exercise #18

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Exercise #19

Video Solution

Answer

Exercise #20

Video Solution

Answer