Question

name: kyree wegiele date: laws of exponents: negative rule the rule when you have a negative exponent, take the reciprocal of the base and exponent. (you never want a negative exponent in an expression.) $a^{-m}=\frac{1}{a^{m}}$ when $a \
eq 0$ examples a.) $x^{-3}=\frac{1}{x^{3}}$ b.) $x^{-5}y^{2}=\frac{y^{2}}{x^{5}}$ c.) $\frac{1}{y^{-8}}=y^{8}$ d.) $x^{-4}x^{3}=x^{-1}=\frac{1}{x}$ *apply the product rule first. practice simplify each expression using exponential notation. 1.) $x^{-7}$ 2.) $\frac{1}{y^{-1}}$ 3.) $a^{-8}b^{3}$ 4.) $3m^{4}n^{-12}$ 5.) $\frac{x^{-6}}{y^{-3}}$ 6.) $\frac{1}{a^{14}b^{-10}}$ 7.) $\frac{4}{m^{-6}n^{-2}}$ 8.) $w^{-3}x^{2}y^{-9}$ 9.) $12a^{-1}a^{3}$ 10.) $(x^{-3}y^{2})(x^{-1}y^{-4})$ 11.) $\frac{m^{-3}n^{4}}{m^{7}n^{-8}}$ 12.) $\frac{2a^{-5}}{b^{6}} \cdot \frac{a}{4b^{2}}$

Explanation:

Step1: Simplify \( x^{-7} \)

Using the negative exponent rule \( a^{-m}=\frac{1}{a^{m}} \), for \( a = x \) and \( m = 7 \), we get \( x^{-7}=\frac{1}{x^{7}} \).

Step2: Simplify \( \frac{1}{y^{-1}} \)

Using the rule \( \frac{1}{a^{-m}}=a^{m} \) (since \( a^{-m}=\frac{1}{a^{m}} \), taking reciprocal gives \( a^{m} \)), for \( a = y \) and \( m = 1 \), we have \( \frac{1}{y^{-1}}=y^{1}=y \).

Step3: Simplify \( a^{-8}b^{3} \)

For the term with negative exponent \( a^{-8} \), apply \( a^{-m}=\frac{1}{a^{m}} \), so \( a^{-8}b^{3}=\frac{b^{3}}{a^{8}} \).

Step4: Simplify \( 3m^{4}n^{-12} \)

For \( n^{-12} \), use \( a^{-m}=\frac{1}{a^{m}} \), so \( 3m^{4}n^{-12}=\frac{3m^{4}}{n^{12}} \).

Step5: Simplify \( \frac{x^{-6}}{y^{-3}} \)

Apply the negative exponent rule to both numerator and denominator. \( x^{-6}=\frac{1}{x^{6}} \) and \( y^{-3}=\frac{1}{y^{3}} \), so \( \frac{x^{-6}}{y^{-3}}=\frac{\frac{1}{x^{6}}}{\frac{1}{y^{3}}}=\frac{y^{3}}{x^{6}} \) (dividing by a fraction is multiplying by its reciprocal).

Step6: Simplify \( \frac{1}{a^{14}b^{-10}} \)

For \( b^{-10} \), \( \frac{1}{b^{-10}} = b^{10} \), so \( \frac{1}{a^{14}b^{-10}}=\frac{b^{10}}{a^{14}} \).

Step7: Simplify \( \frac{4}{m^{-6}n^{-2}} \)

For \( m^{-6} \), \( \frac{1}{m^{-6}} = m^{6} \); for \( n^{-2} \), \( \frac{1}{n^{-2}} = n^{2} \). So \( \frac{4}{m^{-6}n^{-2}} = 4m^{6}n^{2} \).

Step8: Simplify \( w^{-3}x^{2}y^{-9} \)

Apply negative exponent rule to \( w^{-3} \) and \( y^{-9} \): \( w^{-3}=\frac{1}{w^{3}} \), \( y^{-9}=\frac{1}{y^{9}} \), so \( w^{-3}x^{2}y^{-9}=\frac{x^{2}}{w^{3}y^{9}} \).

Step9: Simplify \( 12a^{-1}a^{3} \)

First, use the product rule for exponents \( a^{m} \cdot a^{n}=a^{m + n} \) on \( a^{-1}a^{3} \): \( a^{-1 + 3}=a^{2} \). Then multiply by 12: \( 12a^{2} \).

Step10: Simplify \( (x^{-3}y^{2})(x^{-1}y^{-4}) \)

Use product rule for exponents on \( x \) terms: \( x^{-3} \cdot x^{-1}=x^{-3 + (-1)}=x^{-4} \); on \( y \) terms: \( y^{2} \cdot y^{-4}=y^{2 + (-4)}=y^{-2} \). Now apply negative exponent rule: \( x^{-4}y^{-2}=\frac{1}{x^{4}y^{2}} \).

Step11: Simplify \( \frac{m^{-3}n^{4}}{m^{7}n^{-8}} \)

Use quotient rule for exponents \( \frac{a^{m}}{a^{n}}=a^{m - n} \) on \( m \) terms: \( m^{-3 - 7}=m^{-10} \); on \( n \) terms: \( n^{4 - (-8)}=n^{12} \). Then apply negative exponent rule to \( m^{-10} \): \( \frac{m^{-10}n^{12}}{1}=\frac{n^{12}}{m^{10}} \).

Step12: Simplify \( \frac{2a^{-5}}{b^{6}} \cdot \frac{a}{4b^{2}} \)

First, multiply the numerators and denominators: \( \frac{2a^{-5} \cdot a}{b^{6} \cdot 4b^{2}} \). Use product rule on \( a \) terms: \( a^{-5 + 1}=a^{-4} \); on \( b \) terms: \( b^{6 + 2}=b^{8} \). Also, simplify the coefficients: \( \frac{2}{4}=\frac{1}{2} \). Now we have \( \frac{1 \cdot a^{-4}}{2 \cdot b^{8}} \). Apply negative exponent rule to \( a^{-4} \): \( \frac{1}{2a^{4}b^{8}} \).

Answer:

1. \( \boldsymbol{\frac{1}{x^7}} \)
2. \( \boldsymbol{y} \)
3. \( \boldsymbol{\frac{b^3}{a^8}} \)
4. \( \boldsymbol{\frac{3m^4}{n^{12}}} \)
5. \( \boldsymbol{\frac{y^3}{x^6}} \)
6. \( \boldsymbol{\frac{b^{10}}{a^{14}}} \)
7. \( \boldsymbol{4m^6n^2} \)
8. \( \boldsymbol{\frac{x^2}{w^3y^9}} \)
9. \( \boldsymbol{12a^2} \)
10. \( \boldsymbol{\frac{1}{x^4y^2}} \)
11. \( \boldsymbol{\frac{n^{12}}{m^{10}}} \)
12. \( \boldsymbol{\frac{1}{2a^4b^8}} \)