### Home > INT3 > Chapter 4 > Lesson 4.4.1 > Problem4-108

4-108.

Find values of $a$ and $b$ that make each system of equations true (i.e., solve each system). Be sure to show your work or explain your thinking clearly.

1.  $6=a·b^0$
$24=a·b^2$

Remember that $b^0=1$.

You can use this fact to make the first equation $6=a$, and then substitute $6$ for $a$ in the second equation.

Solve for $b$.

$24 = 6b^2$
$4=b^2$
$b = ±2$

$a=6,\ b=±2$

1.  $32=a·b^2$
$128=a·b^3$

Here are three possible methods for solving.

Since $ab^3 = (ab^2)b$, you can use substitution.
Then solve for $b$ and substitute to find $a$.

$128 = (ab^2)b$ or $128 = 32b$

Solve the first equation for $a$ and substitute into the second equation. Then solve for $b$ and substitute to find $a$.

$\frac{32}{b^2}=a$
$128=\frac{32}{b^2}(b^3)$

Since neither $a$ nor $b$ can equal $0$, divide the second by the first.

$\frac{128=ab^3}{32=ab^2}$

$b=4$

$a=2,\ b=4$