Math / Algebra

Question11. Yhat is the result of expression $(5 \sqrt{7})^{2}$ ? A) 125 (B) 175 C) 245 D) 375
12. What is the simplest form of the expression $\sqrt{a^{2} b^{10}}$ ? $|a|\left|b^{5}\right|$ A) $|a| b^{5}$ B) $a b^{5}$ C) $\left|a b^{5}\right|$ D) $a\left|b^{5}\right|$

Studdy Solution

STEP 1

What is this asking? We need to simplify two expressions, one involving a square root and a power, and the other involving the square root of variables. Watch out! Remember the rules of exponents and square roots, especially when dealing with variables!
Don't forget those absolute value signs when taking square roots of variables with even exponents.

STEP 2

1. Simplify the first expression
2. Simplify the second expression

STEP 3

Alright, let's tackle this first expression, $(5 \sqrt{7})^{2}$ !
Remember, squaring something means multiplying it by itself.
So, let's rewrite it:
$(5 \sqrt{7})^{2} = (5 \sqrt{7}) \cdot (5 \sqrt{7})$

STEP 4

Now, we can rearrange things a bit, grouping the numbers and the square roots together:
$(5 \cdot 5) \cdot (\sqrt{7} \cdot \sqrt{7})$

STEP 5

Let's multiply those numbers: $5 \cdot 5 = 25$ .
And remember, multiplying a square root by itself is like undoing the square root!
So, $\sqrt{7} \cdot \sqrt{7} = 7$ .
This gives us:
$25 \cdot 7$

STEP 6

Finally, let's multiply these together to get our final answer:
$25 \cdot 7 = 175$

STEP 7

Now, for the second expression, $\sqrt{a^{2} b^{10}}$ .
We can rewrite this using the properties of square roots:
$\sqrt{a^{2} b^{10}} = \sqrt{a^{2}} \cdot \sqrt{b^{10}}$

STEP 8

Remember, the square root of a variable raised to an even power is the absolute value of the variable raised to half that power.
So, $\sqrt{a^{2}} = |a|$ .
And $\sqrt{b^{10}} = |b^{5}|$ .
This gives us:
$|a| \cdot |b^{5}|$

STEP 9

Since $|b^5|$ is always non-negative, we can rewrite this as:
$|a| \cdot |b^{5}| = |a| b^{5}$

STEP 10

So our final simplified expression is:
$|a| b^{5}$

STEP 11

The simplified form of $(5 \sqrt{7})^{2}$ is 175 (B).
The simplest form of $\sqrt{a^{2} b^{10}}$ is $|a| b^{5}$ (A).

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Question11. Yhat is the result of expression $(5 \sqrt{7})^{2}$ ? A) 125 (B) 175 C) 245 D) 375
12. What is the simplest form of the expression $\sqrt{a^{2} b^{10}}$ ? $|a|\left|b^{5}\right|$ A) $|a| b^{5}$ B) $a b^{5}$ C) $\left|a b^{5}\right|$ D) $a\left|b^{5}\right|$

Studdy Solution

STEP 1

STEP 2

1. Simplify the first expression
2. Simplify the second expression

STEP 3

Alright, let's tackle this first expression, $(5 \sqrt{7})^{2}$ !
Remember, squaring something means multiplying it by itself.
So, let's rewrite it:
$(5 \sqrt{7})^{2} = (5 \sqrt{7}) \cdot (5 \sqrt{7})$

STEP 4

Now, we can rearrange things a bit, grouping the numbers and the square roots together:
$(5 \cdot 5) \cdot (\sqrt{7} \cdot \sqrt{7})$

STEP 5

Let's multiply those numbers: $5 \cdot 5 = 25$ .
And remember, multiplying a square root by itself is like undoing the square root!
So, $\sqrt{7} \cdot \sqrt{7} = 7$ .
This gives us:
$25 \cdot 7$

STEP 6

Finally, let's multiply these together to get our final answer:
$25 \cdot 7 = 175$

STEP 7

Now, for the second expression, $\sqrt{a^{2} b^{10}}$ .
We can rewrite this using the properties of square roots:
$\sqrt{a^{2} b^{10}} = \sqrt{a^{2}} \cdot \sqrt{b^{10}}$

STEP 8

Remember, the square root of a variable raised to an even power is the absolute value of the variable raised to half that power.
So, $\sqrt{a^{2}} = |a|$ .
And $\sqrt{b^{10}} = |b^{5}|$ .
This gives us:
$|a| \cdot |b^{5}|$

STEP 9

Since $|b^5|$ is always non-negative, we can rewrite this as:
$|a| \cdot |b^{5}| = |a| b^{5}$

STEP 10

So our final simplified expression is:
$|a| b^{5}$

STEP 11

The simplified form of $(5 \sqrt{7})^{2}$ is 175 (B).
The simplest form of $\sqrt{a^{2} b^{10}}$ is $|a| b^{5}$ (A).

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Studdy Solution

STEP 1

STEP 2

1. Simplify the first expression2. Simplify the second expression

STEP 3

Alright, let's **tackle** this first expression, (57)2(5 \sqrt{7})^{2}(57​)2!Remember, squaring something means multiplying it by itself.So, let's **rewrite** it:(57)2=(57)⋅(57)(5 \sqrt{7})^{2} = (5 \sqrt{7}) \cdot (5 \sqrt{7})(57​)2=(57​)⋅(57​)

STEP 4

Now, we can **rearrange** things a bit, grouping the numbers and the square roots together:(5⋅5)⋅(7⋅7)(5 \cdot 5) \cdot (\sqrt{7} \cdot \sqrt{7})(5⋅5)⋅(7​⋅7​)

STEP 5

Let's **multiply** those numbers: 5⋅5=255 \cdot 5 = 255⋅5=25.And remember, multiplying a square root by itself is like **undoing** the square root!So, 7⋅7=7\sqrt{7} \cdot \sqrt{7} = 77​⋅7​=7.This gives us:25⋅725 \cdot 725⋅7

STEP 6

Finally, let's **multiply** these together to get our **final answer**:25⋅7=17525 \cdot 7 = 17525⋅7=175

STEP 7

Now, for the second expression, a2b10\sqrt{a^{2} b^{10}}a2b10​.We can **rewrite** this using the **properties of square roots**:a2b10=a2⋅b10\sqrt{a^{2} b^{10}} = \sqrt{a^{2}} \cdot \sqrt{b^{10}}a2b10​=a2​⋅b10​

STEP 8

Remember, the square root of a variable raised to an even power is the absolute value of the variable raised to half that power.So, a2=∣a∣\sqrt{a^{2}} = |a|a2​=∣a∣.And b10=∣b5∣\sqrt{b^{10}} = |b^{5}|b10​=∣b5∣.This gives us:∣a∣⋅∣b5∣|a| \cdot |b^{5}|∣a∣⋅∣b5∣

STEP 9

Since ∣b5∣|b^5|∣b5∣ is always non-negative, we can **rewrite** this as:∣a∣⋅∣b5∣=∣a∣b5|a| \cdot |b^{5}| = |a| b^{5}∣a∣⋅∣b5∣=∣a∣b5

STEP 10

So our **final simplified expression** is:∣a∣b5|a| b^{5}∣a∣b5

STEP 11

The simplified form of (57)2(5 \sqrt{7})^{2}(57​)2 is **175 (B)**.The simplest form of a2b10\sqrt{a^{2} b^{10}}a2b10​ is ∣a∣b5|a| b^{5}∣a∣b5 **(A)**.

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1. Simplify the first expression
2. Simplify the second expression

Alright, let's tackle this first expression, $(5 \sqrt{7})^{2}$ !
Remember, squaring something means multiplying it by itself.
So, let's rewrite it:
$(5 \sqrt{7})^{2} = (5 \sqrt{7}) \cdot (5 \sqrt{7})$

Now, we can rearrange things a bit, grouping the numbers and the square roots together:
$(5 \cdot 5) \cdot (\sqrt{7} \cdot \sqrt{7})$

Let's multiply those numbers: $5 \cdot 5 = 25$ .
And remember, multiplying a square root by itself is like undoing the square root!
So, $\sqrt{7} \cdot \sqrt{7} = 7$ .
This gives us:
$25 \cdot 7$

Finally, let's multiply these together to get our final answer:
$25 \cdot 7 = 175$

Now, for the second expression, $\sqrt{a^{2} b^{10}}$ .
We can rewrite this using the properties of square roots:
$\sqrt{a^{2} b^{10}} = \sqrt{a^{2}} \cdot \sqrt{b^{10}}$

Remember, the square root of a variable raised to an even power is the absolute value of the variable raised to half that power.
So, $\sqrt{a^{2}} = |a|$ .
And $\sqrt{b^{10}} = |b^{5}|$ .
This gives us:
$|a| \cdot |b^{5}|$

Since $|b^5|$ is always non-negative, we can rewrite this as:
$|a| \cdot |b^{5}| = |a| b^{5}$

So our final simplified expression is:
$|a| b^{5}$

The simplified form of $(5 \sqrt{7})^{2}$ is 175 (B).
The simplest form of $\sqrt{a^{2} b^{10}}$ is $|a| b^{5}$ (A).