Simplifying Radicals - Kuta Software - Free Printable

Here are the step-by-step solutions for each problem on the worksheet.

1) $\sqrt{24}$
* Find perfect square factors of 24: $24 = 4 \times 6$.
* Take the square root of 4, which is 2.
* Leave the 6 inside.
* Answer: $2\sqrt{6}$

2) $\sqrt[3]{1000}$
* Find a number that equals 1000 when multiplied by itself three times.
* $10 \times 10 \times 10 = 1000$.
* Answer: $10$

3) $\sqrt[5]{-162}$
* Since the index (5) is odd, the answer will be negative.
* Ignore the sign for a moment and look at 162.
* $162 = 2 \times 81 = 2 \times 3^4$. There are no groups of five identical factors.
* It cannot be simplified further.
* Answer: $\sqrt[5]{-162}$ or $-\sqrt[5]{162}$

4) $\sqrt{512}$
* Find perfect square factors. $512 = 256 \times 2$.
* $\sqrt{256} = 16$.
* Answer: $16\sqrt{2}$

5) $\sqrt[4]{128n^4}$
* Break down 128: $128 = 16 \times 8 = 2^4 \times 8$. The fourth root of $2^4$ is 2.
* Break down $n^4$: The fourth root of $n^4$ is $|n|$ (absolute value because the index is even).
* Answer: $2|n|\sqrt[4]{8}$

6) $\sqrt{98k}$
* Break down 98: $98 = 49 \times 2$.
* $\sqrt{49} = 7$.
* Answer: $7\sqrt{2k}$

7) $\sqrt[4]{224r^7}$
* Break down 224: $224 = 16 \times 14 = 2^4 \times 14$. Fourth root of $2^4$ is 2.
* Break down $r^7$: $r^7 = r^4 \times r^3$. Fourth root of $r^4$ is $|r|$.
* Answer: $2|r|\sqrt[4]{14r^3}$

8) $\sqrt[3]{24m^5}$
* Break down 24: $24 = 8 \times 3 = 2^3 \times 3$. Cube root of $2^3$ is 2.
* Break down $m^5$: $m^5 = m^3 \times m^2$. Cube root of $m^3$ is $m$.
* Answer: $2m\sqrt[3]{3m^2}$

9) $\sqrt{392x^2}$
* Break down 392: $392 = 196 \times 2$. $\sqrt{196} = 14$.
* Break down $x^2$: $\sqrt{x^2} = |x|$ (absolute value because it's an even root).
* Answer: $14|x|\sqrt{2}$

10) $\sqrt[4]{512x^2}$
* Break down 512: $512 = 256 \times 2 = 4^4 \times 2$. Fourth root of $4^4$ is 4.
* $x^2$ stays inside because the power (2) is less than the index (4).
* Answer: $4\sqrt[4]{2x^2}$

11) $\sqrt[4]{405x^3y^2}$
* Break down 405: $405 = 81 \times 5 = 3^4 \times 5$. Fourth root of $3^4$ is 3.
* $x^3$ and $y^2$ stay inside because their powers are less than 4.
* Answer: $3\sqrt[4]{5x^3y^2}$

12) $\sqrt[5]{-16a^3b^8}$
* Negative sign stays outside (odd root).
* Break down 16: No fifth powers ($2^4=16$, not enough).
* Break down $a^3$: Stays inside.
* Break down $b^8$: $b^8 = b^5 \times b^3$. Fifth root of $b^5$ is $b$.
* Answer: $-b\sqrt[5]{16a^3b^3}$

13) $\sqrt[4]{128x^5y^7}$
* Break down 128: $128 = 16 \times 8 = 2^4 \times 8$. Fourth root of $2^4$ is 2.
* Break down $x^5$: $x^4 \times x$. Fourth root of $x^4$ is $|x|$.
* Break down $y^7$: $y^4 \times y^3$. Fourth root of $y^4$ is $|y|$.
* Answer: $2|x||y|\sqrt[4]{8xy^3}$

14) $\sqrt[4]{16xy}$
* Break down 16: $16 = 2^4$. Fourth root is 2.
* $x$ and $y$ stay inside.
* Answer: $2\sqrt[4]{xy}$

15) $\sqrt[4]{448x^4y^7}$
* Break down 448: $448 = 16 \times 28 = 2^4 \times 28$. Fourth root of $2^4$ is 2.
* Break down $x^4$: Fourth root is $|x|$.
* Break down $y^7$: $y^4 \times y^3$. Fourth root of $y^4$ is $|y|$.
* Answer: $2|x||y|\sqrt[4]{28y^3}$

16) $\sqrt[5]{56x^5y}$
* Break down 56: $56 = 8 \times 7$. No fifth powers.
* Break down $x^5$: Fifth root is $x$ (odd index, so no absolute value needed).
* $y$ stays inside.
* Answer: $x\sqrt[5]{56y}$

17) What simplifies into $2mn^2\sqrt[3]{5mn^2}$?
* To find the original expression, move everything back inside the cube root by cubing it.
* Outside part: $(2mn^2)^3 = 2^3 \cdot m^3 \cdot (n^2)^3 = 8m^3n^6$.
* Inside part: $5mn^2$.
* Multiply them: $8m^3n^6 \cdot 5mn^2 = 40m^{3+1}n^{6+2} = 40m^4n^8$.
* Answer: $\sqrt[3]{40m^4n^8}$

18) Simplify $\sqrt[6]{3 \cdot 2^6 \cdot x^{26}y^{x+3}}$
* Index is 6. We look for powers of 6.
* $2^6$: Sixth root is 2.
* $x^{26}$: Closest multiple of 6 below 26 is 24. $x^{26} = x^{24} \cdot x^2$. Sixth root of $x^{24}$ is $x^4$ ($24/6=4$). Since the index is even, we use absolute value: $|x^4|$ which is just $x^4$ because any real number to the 4th power is positive.
* $y^{x+3}$: We don't know what $x$ is, so we assume the exponent isn't a multiple of 6. It stays inside.
* 3 stays inside.
* $x^2$ stays inside.
* Answer: $2x^4\sqrt[6]{3x^2y^{x+3}}$

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Final Answer:
1) $2\sqrt{6}$
2) $10$
3) $\sqrt[5]{-162}$
4) $16\sqrt{2}$
5) $2|n|\sqrt[4]{8}$
6) $7\sqrt{2k}$
7) $2|r|\sqrt[4]{14r^3}$
8) $2m\sqrt[3]{3m^2}$
9) $14|x|\sqrt{2}$
10) $4\sqrt[4]{2x^2}$
11) $3\sqrt[4]{5x^3y^2}$
12) $-b\sqrt[5]{16a^3b^3}$
13) $2|x||y|\sqrt[4]{8xy^3}$
14) $2\sqrt[4]{xy}$
15) $2|x||y|\sqrt[4]{28y^3}$
16) $x\sqrt[5]{56y}$
17) $\sqrt[3]{40m^4n^8}$
18) $2x^4\sqrt[6]{3x^2y^{x+3}}$