Question

exponents & radicals practice
date 1/15/20
simplify. your answer should contain only positive exponents.

1. $2v^{3} \cdot 3v^{2}$
2. $2x^{-1}y^{-3} \cdot 4x^{2} \cdot 3yx^{3}$
3. $4yz^{-1} \cdot 4x^{2}y^{-1}z^{4} \cdot 3xy^{3}$
4. $4pm^{4}q^{3} \cdot 4m^{4}p^{3}q^{4}$
5. $(k^{-4})^{-1}$
6. $(a^{-2}b^{0})^{-2}$
7. $(3x^{2}y^{0})^{0}$
8. $(2a^{3}b^{4}c^{0})^{2}$
9. $\frac{n^{-1}}{2n^{4}}$
10. $\frac{4a^{-3}b^{-1}}{4b^{2}}$
11. $\frac{3zx^{-4}y^{4}}{3y^{-2}z^{3}}$
12. $\frac{3m^{-1}n^{-2}p^{2}}{3m^{-3}p^{3}}$

Explanation:

Step1: Multiply coefficients, add exponents

$2 \cdot 3 \cdot v^{3+2} = 6v^5$

Step2: Multiply coefficients, add exponents

$2 \cdot 4 \cdot 3 \cdot x^{-1+2+3} \cdot y^{-3+1} = 24x^4y^{-2} = \frac{24x^4}{y^2}$

Step3: Multiply coefficients, add exponents

$4 \cdot 4 \cdot 3 \cdot x^{2+1} \cdot y^{1-1+3} \cdot z^{-1+4} = 48x^3y^3z^3$

Step4: Multiply coefficients, add exponents

$4 \cdot 4 \cdot p^{1+1} \cdot m^{4+4} \cdot q^{3+4} = 16p^2m^8q^7$

Step5: Apply power of a power rule

$k^{(-4) \cdot (-1)} = k^4$

Step6: Simplify $b^0$, apply power rule

$b^0=1$, so $(a^{-2})^{-2} = a^{(-2) \cdot (-2)} = a^4$

Step7: Apply zero exponent rule

Any non-zero term to 0 power is 1: $1$

Step8: Simplify $c^0$, apply power rule

$c^0=1$, so $2^2 \cdot a^{3 \cdot 2} \cdot b^{4 \cdot 2} = 4a^6b^8$

Step9: Combine exponents, rewrite positive

$\frac{1}{2}n^{-1-4} = \frac{1}{2n^5}$

Step10: Cancel constants, add exponents

$\frac{4}{4}a^{-3}b^{-1-2} = \frac{1}{a^3b^3}$

Step11: Cancel constants, add exponents

$\frac{3}{3}x^{-4}y^{4-(-2)}z^{1-3} = \frac{y^6}{x^4z^2}$

Step12: Cancel constants, add exponents

$\frac{3}{3}m^{-1-(-3)}n^{-2}p^{2-3} = \frac{m^2}{n^2p}$

Answer:

1. $6v^5$
2. $\frac{24x^4}{y^2}$
3. $48x^3y^3z^3$
4. $16p^2m^8q^7$
5. $k^4$
6. $a^4$
7. $1$
8. $4a^6b^8$
9. $\frac{1}{2n^5}$
10. $\frac{1}{a^3b^3}$
11. $\frac{y^6}{x^4z^2}$
12. $\frac{m^2}{n^2p}$