Question

1. reinforce simplify each numerical expression using the laws of exponents. leave your answers in exponential form.

a. ( 5^{3} cdot 5^{4}= )____ b. ( 3^{6} cdot 4^{3} cdot 3^{-2}= )__ c. ( \frac{x^{5}}{x^{-2}}= )____
d. ( \frac{3 x^{5}}{y^{4}} cdotleft(\frac{3^{3} x^{-3}}{y^{2}}
ight)^{-2}= )____ e. ( \frac{12 x^{3} y^{-6}}{z^{4}} cdot \frac{x^{2} y^{3}}{4 z^{-10}}= )____ f. ( left(5 x^{-2} y^{5}
ight)^{3}= )______

Explanation:

Step1: Simplify \(5^{3}\cdot5^{4}\)

Using the law \(a^{m}\cdot a^{n}=a^{m + n}\), we have \(5^{3+4}=5^{7}\)

Step2: Simplify \(3^{6}\cdot4^{3}\cdot3^{-2}\)

Using \(a^{m}\cdot a^{n}=a^{m + n}\) for the terms with base \(3\): \(3^{6+( - 2)}\cdot4^{3}=3^{4}\cdot4^{3}\)

Step3: Simplify \(\frac{x^{5}}{x^{-2}}\)

Using the law \(\frac{a^{m}}{a^{n}}=a^{m - n}\), we get \(x^{5-( - 2)}=x^{7}\)

Step4: Simplify \(\frac{3x^{5}}{y^{4}}\cdot(\frac{3^{3}x^{-3}}{y^{2}})^{-2}\)

First, \((\frac{3^{3}x^{-3}}{y^{2}})^{-2}=\frac{3^{-6}x^{6}}{y^{-4}}\) (using \((\frac{a}{b})^{-n}=\frac{b^{n}}{a^{n}}\) and \((a^{m})^{n}=a^{mn}\)). Then \(\frac{3x^{5}}{y^{4}}\cdot\frac{3^{-6}x^{6}}{y^{-4}}=\frac{3^{1+( - 6)}x^{5 + 6}}{y^{4+( - 4)}}=\frac{3^{-5}x^{11}}{y^{0}}=\frac{x^{11}}{3^{5}}\) (since \(y^{0}=1\))

Step5: Simplify \(\frac{12x^{3}y^{-6}}{z^{4}}\cdot\frac{x^{2}y^{3}}{4z^{-10}}\)

First, \(\frac{12}{4}=3\). Then using \(a^{m}\cdot a^{n}=a^{m + n}\) for \(x\): \(x^{3+2}=x^{5}\), for \(y\): \(y^{-6 + 3}=y^{-3}\), for \(z\): \(z^{-4-( - 10)}=z^{6}\). So the result is \(3x^{5}y^{-3}z^{6}=\frac{3x^{5}z^{6}}{y^{3}}\)

Step6: Simplify \((5x^{-2}y^{5})^{3}\)

Using \((ab)^{n}=a^{n}b^{n}\) and \((a^{m})^{n}=a^{mn}\), we get \(5^{3}x^{-6}y^{15}=\frac{125y^{15}}{x^{6}}\)

Answer:

a. \(5^{7}\)
b. \(3^{4}\cdot4^{3}\)
c. \(x^{7}\)
d. \(\frac{x^{11}}{3^{5}}\)
e. \(\frac{3x^{5}z^{6}}{y^{3}}\)
f. \(\frac{125y^{15}}{x^{6}}\)