QuestionSimplify using absolute value signs as needed. (a) $\sqrt[4]{405 m^{10}}$ $\square$ (b) $\sqrt[5]{160 n^{8}}$ $\square$

Studdy Solution

STEP 1

What is this asking? We need to simplify two expressions involving radicals, and we might need absolute values depending on the index of the radical. Watch out! Remember that even roots of variables might need absolute value signs to ensure we get a non-negative result, while odd roots don't have this requirement!

STEP 2

1. Simplify the fourth root
2. Simplify the fifth root

STEP 3

Alright, let's break down $\sqrt[4]{405 m^{10}}$ !
First, we're dealing with a fourth root, which means an even index, so keep those absolute values in mind!

STEP 4

Let's factor out the number part: $405 = 3 \cdot 3 \cdot 3 \cdot 3 \cdot 5 = 3^4 \cdot 5 = 81 \cdot 5$ So, we can rewrite the expression as: $\sqrt[4]{81 \cdot 5 \cdot m^{10}}$

STEP 5

Now, let's tackle the variable part, $m^{10}$ .
Since $10 = 4 \cdot 2 + 2$ , we can rewrite $m^{10}$ as $m^{4 \cdot 2 + 2} = (m^4)^2 \cdot m^2$ .
This is helpful because we're taking the fourth root!

STEP 6

Let's put it all together: $\sqrt[4]{81 \cdot 5 \cdot m^{10}} = \sqrt[4]{3^4 \cdot 5 \cdot (m^4)^2 \cdot m^2} = \sqrt[4]{3^4} \cdot \sqrt[4]{(m^4)^2} \cdot \sqrt[4]{5m^2} = 3 \cdot m^2 \cdot \sqrt[4]{5m^2}$ Since we're taking an even root and $m$ could be negative, we need the absolute value around $m^2$ , which gives us $|m^2|$ .
But wait, $m^2$ is always non-negative, so $|m^2| = m^2$ !

STEP 7

Our simplified expression is $3m^2\sqrt[4]{5m^2}$ .
Awesome!

STEP 8

Time to simplify $\sqrt[5]{160n^8}$ !
This time, we have an odd index, so no need to worry about absolute values!

STEP 9

Let's factor the number: $160 = 2 \cdot 2 \cdot 2 \cdot 2 \cdot 2 \cdot 5 = 2^5 \cdot 5 = 32 \cdot 5$ So, we have $\sqrt[5]{32 \cdot 5 \cdot n^8}$ .

STEP 10

Now, for the variable part: $8 = 5 \cdot 1 + 3$ , so $n^8 = n^{5+3} = n^5 \cdot n^3$ .

STEP 11

Let's combine everything: $\sqrt[5]{32 \cdot 5 \cdot n^8} = \sqrt[5]{2^5 \cdot 5 \cdot n^5 \cdot n^3} = \sqrt[5]{2^5} \cdot \sqrt[5]{n^5} \cdot \sqrt[5]{5n^3} = 2 \cdot n \cdot \sqrt[5]{5n^3} = 2n\sqrt[5]{5n^3}$

STEP 12

Our simplified expression is $2n\sqrt[5]{5n^3}$ .
Fantastic!

STEP 13

(a) $3m^2\sqrt[4]{5m^2}$ (b) $2n\sqrt[5]{5n^3}$

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