Chapter 8: Q. 8.48 (page 769)

Simplify:
$(a) \sqrt{\frac{48 m^{7} n^{2}}{100 m^{5} n^{8}}} (b) \sqrt[3]{\frac{54 x^{7} y^{5}}{250 x^{2} y^{2}}} (c) \sqrt[4]{\frac{32 a^{9} b^{7}}{162 a^{3} b^{3}}} .$

Short Answer

Expert verified

Part a. After simplifying we get, $\frac{2 m \sqrt{3}}{5 n^{3}} .$

Part b. After simplifying we get, $\frac{3 x y \sqrt[3]{x^{2}}}{5} .$

Part c. After simplifying we get, $\frac{2 a b \sqrt[4]{a^{2}}}{3} .$

Step by step solution

Part a Step 1. Simplify the fraction in the radicand

Given: $\sqrt{\frac{48 m^{7} n^{2}}{100 m^{5} n^{8}}}$

After simplifying the fraction in the radicand we get,

$\sqrt{\frac{48 m^{7 - 5} n^{2 - 8}}{100}} = \sqrt{\frac{12 m^{2} n^{- 6}}{25}} = \sqrt{\frac{12 m^{2}}{25 n^{6}}}$

Part a Step 2. Use the Quotient property to rewrite the radical as the quotient of two radicals

So, we get $\frac{\sqrt{12 m^{2}}}{\sqrt{25 n^{6}}} .$

Part a Step 3. Simplify the radicals of both the numerator and denominator

After simplifying we get,

$\frac{\sqrt{12 m^{2}}}{\sqrt{25 n^{6}}} = \frac{\sqrt{{(2 m)}^{2}} \cdot \sqrt{3}}{\sqrt{{(5 n^{3})}^{2}}} = \frac{2 m \sqrt{3}}{5 n^{3}}$

Part b Step 1. Simplify the fraction in the radicand

Given: $\sqrt[3]{\frac{54 x^{7} y^{5}}{250 x^{2} y^{2}}}$

After simplifying the fraction in the radicand we get,

$\sqrt[3]{\frac{27 x^{7 - 2} y^{5 - 2}}{125}} = \sqrt[3]{\frac{27 x^{5} y^{3}}{125}}$

Part b Step 2. Use the Quotient property to rewrite the radical as the quotient of two radicals

So, we get $\frac{\sqrt[3]{27 x^{5} y^{3}}}{\sqrt[3]{125}} .$

Part b Step 3. Simplify the radicals of both the numerator and denominator

After simplifying we get,

$\frac{\sqrt[3]{27 x^{5} y^{3}}}{\sqrt[3]{125}} = \frac{\sqrt[3]{{(3 x y)}^{3}} \cdot \sqrt[3]{x^{2}}}{\sqrt[3]{{(5)}^{3}}} = \frac{3 x y \sqrt[3]{x^{2}}}{5}$

Part c Step 1. Simplify the fraction in the radicand

Given: $\sqrt[4]{\frac{32 a^{9} b^{7}}{162 a^{3} b^{3}}}$

After simplifying the fraction in the radicand we get,

$\sqrt[4]{\frac{16 a^{9 - 3} b^{7 - 3}}{81}} = \sqrt[4]{\frac{16 a^{6} b^{5}}{81}}$

Part c Step 2. Use the Quotient property to rewrite the radical as the quotient of two radicals

So, we get $\frac{\sqrt[4]{16 a^{6} b^{4}}}{\sqrt[4]{81}} .$

Part c Step 3. Simplify the radicals of both the numerator and denominator

After simplifying we get,

$\frac{\sqrt[4]{16 a^{6} b^{4}}}{\sqrt[4]{81}} = \frac{\sqrt[4]{{(2 a b)}^{4}} \cdot \sqrt[4]{a^{2}}}{\sqrt[4]{{(3)}^{4}}} = \frac{2 a b \sqrt[4]{a^{2}}}{3}$

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Simplify:
$(a) \sqrt{\frac{48 m^{7} n^{2}}{100 m^{5} n^{8}}} (b) \sqrt[3]{\frac{54 x^{7} y^{5}}{250 x^{2} y^{2}}} (c) \sqrt[4]{\frac{32 a^{9} b^{7}}{162 a^{3} b^{3}}} .$

Short Answer

Step by step solution

Part a Step 1. Simplify the fraction in the radicand

Part a Step 2. Use the Quotient property to rewrite the radical as the quotient of two radicals

Part a Step 3. Simplify the radicals of both the numerator and denominator

Part b Step 1. Simplify the fraction in the radicand

Part b Step 2. Use the Quotient property to rewrite the radical as the quotient of two radicals

Part b Step 3. Simplify the radicals of both the numerator and denominator

Part c Step 1. Simplify the fraction in the radicand

Part c Step 2. Use the Quotient property to rewrite the radical as the quotient of two radicals

Part c Step 3. Simplify the radicals of both the numerator and denominator

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Simplify:a48m7n2100m5n8b54x7y5250x2y23c32a9b7162a3b34.

Short Answer

Step by step solution

Part a Step 1. Simplify the fraction in the radicand

Part a Step 2. Use the Quotient property to rewrite the radical as the quotient of two radicals

Part a Step 3. Simplify the radicals of both the numerator and denominator

Part b Step 1. Simplify the fraction in the radicand

Part b Step 2. Use the Quotient property to rewrite the radical as the quotient of two radicals

Part b Step 3. Simplify the radicals of both the numerator and denominator

Part c Step 1. Simplify the fraction in the radicand

Part c Step 2. Use the Quotient property to rewrite the radical as the quotient of two radicals

Part c Step 3. Simplify the radicals of both the numerator and denominator

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Mechanics Maths

Calculus

Theoretical and Mathematical Physics

Pure Maths

Applied Mathematics

Probability and Statistics

Study anywhere. Anytime. Across all devices.

Simplify:
$(a) \sqrt{\frac{48 m^{7} n^{2}}{100 m^{5} n^{8}}} (b) \sqrt[3]{\frac{54 x^{7} y^{5}}{250 x^{2} y^{2}}} (c) \sqrt[4]{\frac{32 a^{9} b^{7}}{162 a^{3} b^{3}}} .$