Find the value of "a" which makes this statement true:

Question

The problem is: x^2a-12 ⋅ x^2a+13=x^9a-7

Answer 1

Hello, Angie,

The first expression, x^2a-12 ⋅ x^2a+13 can be simplified. Exponents are summed when multiplied in this manner, so the final exponent is 2a-12+2a+13, or 4a+1. The term on the left side becomes"

x^(4a+1)

The revised equation is now x^(4a+1) = x^(9a-7)

For this equation to be true, both exponents must be the same (i.e., equal to each other). So let's set them equal and solve for a:

4a + 1 = 9a - 7

5a = 8

a = (8/5)

Try the answer in the equation to see if it works:

x^2a-12 ⋅ x^2a+13=x^9a-7

The exponents become, in order that we see them, from left to right. Substitute (8/5) for "a."

x^(2(8/5)-12) ⋅ x^(2(8/5)+13)=x^(9(8/5)a-7)

1) 2a-12 = 2(8/5) - 12, or 16/5 - 12, so the first exponent is = -(44/5). [Ugly]

2) 2a + 13 = 2(8/5) + 13, or 16/5 + 13 so the second exponent is = (81/5) [Ugh]

3) 9a-7 = 9(8/5) - 7, or (72/5)-7 and, drum roll, the right exponent is = (37/5) [Almost there]

Add the exponents on the left side:

(81/5)-(44/5) = (37/5)

We know the right side exponent is (37/5), so the answer a = (8/5) is correct.

I hope this helps. Fun, when it works, and when you keep the parentheses in the correct places. Hard to do, sometimes.

Bob

Answer 2

Hello Angie!

The first step to answering this would be to simplify the left side of the equation. When you multiply variables with exponents together, you add the exponents.

X^2a-12 * X^2a+13 = X^4a+1

Now that the left side is simplified we can rewrite the equation.

X^4a+1 = X^9a-7

Now that we have both sides simplified, we can see that we just need to make the exponents equal to each other.

4a+1 = 9a-7

Next, we need to algebraically solve for a.

First add 7 to both sides.

4a+8 = 9a

Next, subtract 4a from both sides.

8 = 5a

Finally, divide both sides by 5

8/5 = a

or a = 1.6