QuestionSimplify $\sqrt{49 p^{9} n^{4}}$

Studdy Solution

STEP 1

What is this asking? We need to simplify a square root expression with variables raised to powers. Watch out! Remember that the square root of a variable raised to a power is the variable raised to half that power.
Don't forget to simplify the numerical part of the square root too!

STEP 2

1. Simplify the numerical square root.
2. Simplify the square root of the variable terms.
3. Combine the simplified terms.

STEP 3

Alright, let's start with the number inside the square root: $\sqrt{49}$ .
We're looking for a number multiplied by itself that gives us 49.
That's 7, because $7 \cdot 7 = 49$ !
So, $\sqrt{49} = 7$ .

STEP 4

Now, let's look at the p term.
We have $\sqrt{p^9}$ .
Remember, a square root is like raising to the power of $\frac{1}{2}$ .
So, we have $(p^9)^{\frac{1}{2}}$ .
When we raise a power to a power, we multiply the exponents: $9 \cdot \frac{1}{2} = \frac{9}{2}$ .
So, $\sqrt{p^9} = p^{\frac{9}{2}}$ .

STEP 5

We can rewrite $p^{\frac{9}{2}}$ as $p^{4 + \frac{1}{2}} = p^4 \cdot p^{\frac{1}{2}} = p^4\sqrt{p}$ .

STEP 6

Next up is the n term!
We have $\sqrt{n^4}$ .
Similar to what we did with p, we have $(n^4)^{\frac{1}{2}}$ .
Multiplying the exponents gives us $4 \cdot \frac{1}{2} = 2$ .
So, $\sqrt{n^4} = n^2$ .

STEP 7

Time to put it all together!
We found that $\sqrt{49} = 7$ , $\sqrt{p^9} = p^4\sqrt{p}$ , and $\sqrt{n^4} = n^2$ .
Multiplying these simplified terms gives us our final simplified expression: $7p^4n^2\sqrt{p}$ .

STEP 8

The simplified form of $\sqrt{49p^9n^4}$ is $7p^4n^2\sqrt{p}$ .

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