QUESTION IMAGE
Question
non-perfect square roots: use the perfect square numbers below to break the problem apart:
perfect squares:
□ □ □ □ □ □ □ □ □ □, etc....
put the following in simplest radical form:
- √24
2√6
- √48
4√3
- √72
6√2
- √63
- √90
3√10
- √175
5√7
- √162
- √245
- √343
- √117
- √28
- √450
non-perfect cube roots: use the perfect cube numbers below to break the problem apart:
perfect cubes:
□ □ □ □ □ □ □, etc....
To solve these problems, we need to simplify each radical by factoring out the largest perfect square factor from the radicand (the number inside the square root).
Problem 4: $\boldsymbol{\sqrt{63}}$
Step 1: Factor the radicand
We factor 63 to find the largest perfect square factor. We know that $63 = 9 \times 7$, and 9 is a perfect square ($3^2 = 9$).
Step 2: Apply the square root property
Using the property $\sqrt{ab} = \sqrt{a} \times \sqrt{b}$ (where $a \geq 0$ and $b \geq 0$), we can rewrite $\sqrt{63}$ as $\sqrt{9 \times 7} = \sqrt{9} \times \sqrt{7}$.
Step 3: Simplify the perfect square root
Since $\sqrt{9} = 3$, we have $3 \times \sqrt{7} = 3\sqrt{7}$.
Problem 7: $\boldsymbol{\sqrt{162}}$
Step 1: Factor the radicand
Factor 162: $162 = 81 \times 2$, and 81 is a perfect square ($9^2 = 81$).
Step 2: Apply the square root property
$\sqrt{162} = \sqrt{81 \times 2} = \sqrt{81} \times \sqrt{2}$.
Step 3: Simplify the perfect square root
Since $\sqrt{81} = 9$, we get $9 \times \sqrt{2} = 9\sqrt{2}$.
Problem 8: $\boldsymbol{\sqrt{245}}$
Step 1: Factor the radicand
Factor 245: $245 = 49 \times 5$, and 49 is a perfect square ($7^2 = 49$).
Step 2: Apply the square root property
$\sqrt{245} = \sqrt{49 \times 5} = \sqrt{49} \times \sqrt{5}$.
Step 3: Simplify the perfect square root
Since $\sqrt{49} = 7$, we have $7 \times \sqrt{5} = 7\sqrt{5}$.
Problem 9: $\boldsymbol{\sqrt{343}}$
Step 1: Factor the radicand
Factor 343: $343 = 49 \times 7$, and 49 is a perfect square ($7^2 = 49$).
Step 2: Apply the square root property
$\sqrt{343} = \sqrt{49 \times 7} = \sqrt{49} \times \sqrt{7}$.
Step 3: Simplify the perfect square root
Since $\sqrt{49} = 7$, we get $7 \times \sqrt{7} = 7\sqrt{7}$.
Problem 10: $\boldsymbol{\sqrt{117}}$
Step 1: Factor the radicand
Factor 117: $117 = 9 \times 13$, and 9 is a perfect square ($3^2 = 9$).
Step 2: Apply the square root property
$\sqrt{117} = \sqrt{9 \times 13} = \sqrt{9} \times \sqrt{13}$.
Step 3: Simplify the perfect square root
Since $\sqrt{9} = 3$, we have $3 \times \sqrt{13} = 3\sqrt{13}$.
Problem 11: $\boldsymbol{\sqrt{28}}$
Step 1: Factor the radicand
Factor 28: $28 = 4 \times 7$, and 4 is a perfect square ($2^2 = 4$).
Step 2: Apply the square root property
$\sqrt{28} = \sqrt{4 \times 7} = \sqrt{4} \times \sqrt{7}$.
Step 3: Simplify the perfect square root
Since $\sqrt{4} = 2$, we get $2 \times \sqrt{7} = 2\sqrt{7}$.
Problem 12: $\boldsymbol{\sqrt{450}}$
Step 1: Factor the radicand
Factor 450: $450 = 225 \times 2$, and 225 is a perfect square ($15^2 = 225$).
Step 2: Apply the square root property
$\sqrt{450} = \sqrt{225 \times 2} = \sqrt{225} \times \sqrt{2}$.
Step 3: Simplify the perfect square root
Since $\sqrt{225} = 15$, we have $15 \times \sqrt{2} = 15\sqrt{2}$.
Final Answers:
- $\boldsymbol{3\sqrt{7}}$
- $\boldsymbol{9\sqrt{2}}$
- $\boldsymbol{7\sqrt{5}}$
- $\boldsymbol{7\sqrt{7}}$
- $\boldsymbol{3\sqrt{13}}$
- $\boldsymbol{2\sqrt{7}}$
- $\boldsymbol{15\sqrt{2}}$
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To solve these problems, we need to simplify each radical by factoring out the largest perfect square factor from the radicand (the number inside the square root).
Problem 4: $\boldsymbol{\sqrt{63}}$
Step 1: Factor the radicand
We factor 63 to find the largest perfect square factor. We know that $63 = 9 \times 7$, and 9 is a perfect square ($3^2 = 9$).
Step 2: Apply the square root property
Using the property $\sqrt{ab} = \sqrt{a} \times \sqrt{b}$ (where $a \geq 0$ and $b \geq 0$), we can rewrite $\sqrt{63}$ as $\sqrt{9 \times 7} = \sqrt{9} \times \sqrt{7}$.
Step 3: Simplify the perfect square root
Since $\sqrt{9} = 3$, we have $3 \times \sqrt{7} = 3\sqrt{7}$.
Problem 7: $\boldsymbol{\sqrt{162}}$
Step 1: Factor the radicand
Factor 162: $162 = 81 \times 2$, and 81 is a perfect square ($9^2 = 81$).
Step 2: Apply the square root property
$\sqrt{162} = \sqrt{81 \times 2} = \sqrt{81} \times \sqrt{2}$.
Step 3: Simplify the perfect square root
Since $\sqrt{81} = 9$, we get $9 \times \sqrt{2} = 9\sqrt{2}$.
Problem 8: $\boldsymbol{\sqrt{245}}$
Step 1: Factor the radicand
Factor 245: $245 = 49 \times 5$, and 49 is a perfect square ($7^2 = 49$).
Step 2: Apply the square root property
$\sqrt{245} = \sqrt{49 \times 5} = \sqrt{49} \times \sqrt{5}$.
Step 3: Simplify the perfect square root
Since $\sqrt{49} = 7$, we have $7 \times \sqrt{5} = 7\sqrt{5}$.
Problem 9: $\boldsymbol{\sqrt{343}}$
Step 1: Factor the radicand
Factor 343: $343 = 49 \times 7$, and 49 is a perfect square ($7^2 = 49$).
Step 2: Apply the square root property
$\sqrt{343} = \sqrt{49 \times 7} = \sqrt{49} \times \sqrt{7}$.
Step 3: Simplify the perfect square root
Since $\sqrt{49} = 7$, we get $7 \times \sqrt{7} = 7\sqrt{7}$.
Problem 10: $\boldsymbol{\sqrt{117}}$
Step 1: Factor the radicand
Factor 117: $117 = 9 \times 13$, and 9 is a perfect square ($3^2 = 9$).
Step 2: Apply the square root property
$\sqrt{117} = \sqrt{9 \times 13} = \sqrt{9} \times \sqrt{13}$.
Step 3: Simplify the perfect square root
Since $\sqrt{9} = 3$, we have $3 \times \sqrt{13} = 3\sqrt{13}$.
Problem 11: $\boldsymbol{\sqrt{28}}$
Step 1: Factor the radicand
Factor 28: $28 = 4 \times 7$, and 4 is a perfect square ($2^2 = 4$).
Step 2: Apply the square root property
$\sqrt{28} = \sqrt{4 \times 7} = \sqrt{4} \times \sqrt{7}$.
Step 3: Simplify the perfect square root
Since $\sqrt{4} = 2$, we get $2 \times \sqrt{7} = 2\sqrt{7}$.
Problem 12: $\boldsymbol{\sqrt{450}}$
Step 1: Factor the radicand
Factor 450: $450 = 225 \times 2$, and 225 is a perfect square ($15^2 = 225$).
Step 2: Apply the square root property
$\sqrt{450} = \sqrt{225 \times 2} = \sqrt{225} \times \sqrt{2}$.
Step 3: Simplify the perfect square root
Since $\sqrt{225} = 15$, we have $15 \times \sqrt{2} = 15\sqrt{2}$.
Final Answers:
- $\boldsymbol{3\sqrt{7}}$
- $\boldsymbol{9\sqrt{2}}$
- $\boldsymbol{7\sqrt{5}}$
- $\boldsymbol{7\sqrt{7}}$
- $\boldsymbol{3\sqrt{13}}$
- $\boldsymbol{2\sqrt{7}}$
- $\boldsymbol{15\sqrt{2}}$