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Quiz 14
1.) Factor completely: $3a^2b+9ab^2$.
Solution: Since $3$ is in common to both terms, $a$ is in common, and $b$ is in common, we will factor our $3ab$ to get
$$3ab(a+3b).$$
2.) Simplify and write with no negative exponent: $(3a^{-2}b^2)^3$.
Solution: First we use the rule that $(xyz)^n=x^ny^nz^n$ to get
$$3^3 (a^{-2})^3 (b^2)^3.$$
We simplify this using the rule that $(x^n)^m=x^{mn}$ (and for $3^3$ we simply compute $3^3=27$):
$$27 a^{-6} b^6.$$
Now we make the negative exponent go away by recalling that a negative exponent means to put into the denominator of a fraction:
$$\dfrac{27b^6}{a^6}.$$
3.) Write using scientific notation: $0.000000397$.
Solution: $3.97 \times 10^{-7}$
4.) Factor by grouping: $4x^3+2x^2+2x+1$.
Solution: We group the terms and factor them:
$$\begin{array}{ll}
4x^3+2x^2+2x+1 &= (4x^3+2x^2) + (2x+1) \\
&= 2x^2(2x+1) + (2x+1) \\
&= (2x^2+1)(2x+1).
\end{array}$$
