Math  /  Algebra

Question1 2 3 4 5 8 7 \511. 5 11. \square$
Which expression is equivalent to (64y100)12\left(64 y^{100}\right)^{\frac{1}{2}} ? 8y108 y^{10} 8y508 y^{50} 32y1032 y^{10} 32y5032 y^{50} Mark this and return Viewers/AssessmentViewer/Activit.

Studdy Solution

STEP 1

What is this asking? We need to simplify the expression (64y100)12\left(64 y^{100}\right)^{\frac{1}{2}}. Watch out! Remember that a fractional exponent is just another way of writing a root!
Don't mix up the power of yy and the root we're taking.

STEP 2

1. Rewrite the expression using a root.
2. Simplify the constant.
3. Simplify the variable term.
4. Combine the simplified terms.

STEP 3

We're given the expression (64y100)12\left(64 y^{100}\right)^{\frac{1}{2}}.
Remember that raising something to the power of 12\frac{1}{2} is the same as taking the square root!
So, we can **rewrite** our expression as: 64y100 \sqrt{64 y^{100}} This makes it much easier to see what we need to do!

STEP 4

Now, let's **focus** on the **constant term**, 6464.
We need to find the square root of 6464.
What number, multiplied by itself, gives us 6464?
It's 88!
That's because 88=648 \cdot 8 = 64.
So, 64=8\sqrt{64} = 8.
Awesome!

STEP 5

Next up is the **variable term**, y100y^{100}.
We want to find the square root of y100y^{100}, which is the same as (y100)12(y^{100})^{\frac{1}{2}}.
Remember the **power rule**: when you raise a power to a power, you multiply the exponents.

STEP 6

So, we have 10012100 \cdot \frac{1}{2}.
This gives us 1002\frac{100}{2}, which simplifies to **50**.
So, (y100)12=y50(y^{100})^{\frac{1}{2}} = y^{50}.
Fantastic!

STEP 7

We found that 64=8\sqrt{64} = 8 and y100=y50\sqrt{y^{100}} = y^{50}.
Now, we just **put them together**!
So, 64y100\sqrt{64y^{100}} simplifies to 8y508y^{50}.
We did it!

STEP 8

The simplified expression is 8y508y^{50}.

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