Chapter 8: Q. 8.76 (page 791)

Simplify
$\sqrt{27} + 4 \sqrt{3}$
$4 \sqrt[3]{5} - 7 \sqrt[3]{40}$
$\frac{1}{2} \sqrt[3]{128} - \frac{5}{3} \sqrt[3]{54}$

Short Answer

Expert verified

Part a. $7 \sqrt{3}$

Part b. $- 10 \sqrt[3]{5}$

Part c. $- 3 \sqrt[3]{2}$

Step by step solution

Part (a) Step 1. Simplify the radicals and combine like radicals

Using $\sqrt{a b} = \sqrt{a} \cdot \sqrt{b}$ simplify the expression.

$\begin{array}{rcl} \sqrt{27} + 4 \sqrt{3} & = & \sqrt{9 \cdot 3} + 4 \sqrt{3} \\ = & \sqrt{9} \cdot \sqrt{3} + 4 \sqrt{3} \\ = & 3 \sqrt{3} + 4 \sqrt{3} \end{array}$

On combing the like radicals we get

$\begin{array}{rcl} 3 \sqrt{3} + 4 \sqrt{3} & = & (3 + 4) \sqrt{3} \\ = & 7 \sqrt{3} \end{array}$

Part (b) Step 1. Simplify the radicals and combine like radicals

Using $\sqrt{a b} = \sqrt{a} \cdot \sqrt{b}$ simplify the expression.

$\begin{array}{rcl} 4 \sqrt[3]{5} - 7 \sqrt[3]{40} & = & 4 \sqrt[3]{5} - 7 \sqrt[3]{8 \cdot 5} \\ = & 4 \sqrt[3]{5} - 7 \sqrt[3]{8} \cdot \sqrt[3]{5} \\ = & 4 \sqrt[3]{5} - 7 \cdot 2 \cdot \sqrt[3]{5} \\ = & 4 \sqrt[3]{5} - 14 \sqrt[3]{5} \end{array}$

On combing the like radicals we get

$\begin{array}{rcl} 4 \sqrt[3]{5} - 14 \sqrt[3]{5} & = & (4 - 14) \sqrt[3]{5} \\ = & - 10 \sqrt[3]{5} \end{array}$

Part (c) Step 1. Simplify the radicals and combine like radicals

Using $\sqrt{a b} = \sqrt{a} \cdot \sqrt{b}$ simplify the expression.

$\begin{array}{rcl} \frac{1}{2} \sqrt[3]{128} - \frac{5}{3} \sqrt[3]{54} & = & \frac{1}{2} \sqrt[3]{128} - \frac{5}{3} \sqrt[3]{54} \\ = & \frac{1}{2} \sqrt[3]{64 \cdot 2} - \frac{5}{3} \sqrt[3]{27 \cdot 2} \\ = & \frac{1}{2} \sqrt[3]{64} \cdot \sqrt[3]{2} - \frac{5}{3} \sqrt[3]{27} \cdot \sqrt[3]{2} \\ = & \frac{1}{2} \cdot 4 \cdot \sqrt[3]{2} - \frac{5}{3} \cdot 3 \cdot \sqrt[3]{2} \\ = & 2 \sqrt[3]{2} - 5 \sqrt[3]{2} \end{array}$

On combing the like radicals we get

role="math" localid="1644812399604" $\begin{array}{rcl} 2 \sqrt[3]{2} - 5 \sqrt[3]{2} & = & (2 - 5) \sqrt[3]{2} \\ = & - 3 \sqrt[3]{2} \end{array}$

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Simplify
$\sqrt{27} + 4 \sqrt{3}$
$4 \sqrt[3]{5} - 7 \sqrt[3]{40}$
$\frac{1}{2} \sqrt[3]{128} - \frac{5}{3} \sqrt[3]{54}$

Short Answer

Step by step solution

Part (a) Step 1. Simplify the radicals and combine like radicals

Part (b) Step 1. Simplify the radicals and combine like radicals

Part (c) Step 1. Simplify the radicals and combine like radicals

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Simplify27+43453-7403121283-53543

Short Answer

Step by step solution

Part (a) Step 1. Simplify the radicals and combine like radicals

Part (b) Step 1. Simplify the radicals and combine like radicals

Part (c) Step 1. Simplify the radicals and combine like radicals

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Logic and Functions

Geometry

Applied Mathematics

Discrete Mathematics

Probability and Statistics

Decision Maths

Study anywhere. Anytime. Across all devices.

Simplify
$\sqrt{27} + 4 \sqrt{3}$
$4 \sqrt[3]{5} - 7 \sqrt[3]{40}$
$\frac{1}{2} \sqrt[3]{128} - \frac{5}{3} \sqrt[3]{54}$