Math  /  Algebra

QuestionSimplify 24x6y4\sqrt{24 x^{6} y^{-4}}.

Studdy Solution

STEP 1

What is this asking? We need to simplify a square root expression with variables and make it look nicer! Watch out! Remember your exponent rules and don't forget to consider both positive and negative exponents.
Also, watch out for any assumptions about the variables, like whether they're positive or not.

STEP 2

1. Simplify the number part
2. Simplify the xx part
3. Simplify the yy part
4. Combine everything

STEP 3

Let's **break down** 24\sqrt{24}.
We're looking for **perfect squares** hiding inside 24.
We can write 24=4624 = 4 \cdot 6, and since 4=2\sqrt{4} = 2, we get 24=46=46=26\sqrt{24} = \sqrt{4 \cdot 6} = \sqrt{4} \cdot \sqrt{6} = 2\sqrt{6}.
So, we've **simplified** the number part to 262\sqrt{6}.

STEP 4

Now, let's look at x6\sqrt{x^6}.
Remember, a square root is the same as raising to the power of 12\frac{1}{2}.
So, we have (x6)12(x^6)^{\frac{1}{2}}.
Using our **exponent rules**, we multiply the exponents: 612=36 \cdot \frac{1}{2} = 3.
This gives us x3x^3.
Since we're assuming xx is non-negative, we don't need absolute value bars.

STEP 5

Time for the yy part: y4\sqrt{y^{-4}}.
Again, we rewrite the square root as a **fractional exponent**: (y4)12(y^{-4})^{\frac{1}{2}}.
Multiplying the exponents, we get 412=2-4 \cdot \frac{1}{2} = -2.
This gives us y2y^{-2}, which is the same as 1y2\frac{1}{y^2}.
Since we are taking a square root, we assume yy is non-negative, so we don't need absolute value bars.

STEP 6

Let's **put it all together**!
We have 262\sqrt{6}, x3x^3, and 1y2\frac{1}{y^2}.
Multiplying these together gives us our **simplified expression**: 2x36y2.\frac{2x^3\sqrt{6}}{y^2}.

STEP 7

2x36y2\frac{2x^3\sqrt{6}}{y^2}, assuming xx and yy are non-negative.

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