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Chapter 2: Problem 23

Simplify. \(\frac{\left(12 x^{3}\right)^{2}}{\left(10 y^{5}\right)^{2}}\)

Short Answer

Expert verified
The simplified expression is \(\frac{36x^6}{25y^{10}}\).

Step by step solution

01

Apply Exponent Rule

To apply the square to the terms inside the brackets, we need to square each term using the power of a power exponent rule which states that \((a^m)^n = a^{mn}\). In our case, m = 2. \(\frac{(12x^3)^2}{(10y^5)^2} = \frac{12^2(x^3)^2}{10^2(y^5)^2}\)
02

Simplify the Exponents

Now, simplify the exponents by multiplying the powers: \(\frac{12^2(x^3)^2}{10^2(y^5)^2} = \frac{12^2x^{3 \cdot 2}}{10^2y^{5 \cdot 2}}\)
03

Evaluate the Powers

Calculate the squares and multiply the exponents: \(\frac{12^2x^{3 \cdot 2}}{10^2y^{5 \cdot 2}} = \frac{144x^6}{100y^{10}}\)
04

Simplify the Fraction

Notice that both the numerator and denominator have a common factor of 4. So we can simplify the fraction by dividing both by 4: \(\frac{144x^6}{100y^{10}} = \frac{36x^6}{25y^{10}}\) So the simplified expression will be: \[\frac{36x^6}{25y^{10}}\]

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