Question

keep going! use the properties of exponents to simplify the expressions

$15^{-3}=\frac{1}{3375}$

$12^{0}=\square$

$(5^{6})^{3}=\square$

$25^{3}\cdot25^{2}=\square$

$8^{0}=\square$

$\frac{13^{9}}{13^{1}}=\square$

$(36^{3})^{2}=\square$

$\frac{5^{9}}{5^{3}}=\square$

$70^{-6}=\square$

Explanation:

Step1: Recall exponent properties

• Negative exponent: \(a^{-n}=\frac{1}{a^n}\)
• Zero exponent: \(a^0 = 1\) (\(a

eq0\))

• Power of a power: \((a^m)^n=a^{m\times n}\)
• Product of powers: \(a^m\cdot a^n=a^{m + n}\)
• Quotient of powers: \(\frac{a^m}{a^n}=a^{m - n}\)

Step2: Simplify \(12^0\)

Using zero exponent property: \(12^0 = 1\)

Step3: Simplify \((5^6)^3\)

Using power of a power: \((5^6)^3=5^{6\times3}=5^{18}\)

Step4: Simplify \(25^3\cdot25^2\)

Using product of powers: \(25^3\cdot25^2 = 25^{3 + 2}=25^5\) (or \(9765625\) if numerical value is needed)

Step5: Simplify \(8^0\)

Using zero exponent property: \(8^0 = 1\)

Step6: Simplify \(\frac{13^9}{13^1}\)

Using quotient of powers: \(\frac{13^9}{13^1}=13^{9 - 1}=13^8\) (or \(815730721\) if numerical value is needed)

Step7: Simplify \((36^3)^2\)

Using power of a power: \((36^3)^2=36^{3\times2}=36^6\) (or \(2176782336\) if numerical value is needed)

Step8: Simplify \(\frac{5^9}{5^3}\)

Using quotient of powers: \(\frac{5^9}{5^3}=5^{9 - 3}=5^6 = 15625\)

Step9: Simplify \(70^{-6}\)

Using negative exponent property: \(70^{-6}=\frac{1}{70^6}=\frac{1}{117649000000}\)

Answer:

s (filling the boxes):

• \(12^0=\boldsymbol{1}\)
• \((5^6)^3=\boldsymbol{5^{18}}\) (or \(9765625\))
• \(25^3\cdot25^2=\boldsymbol{25^5}\) (or \(9765625\))
• \(8^0=\boldsymbol{1}\)
• \(\frac{13^9}{13^1}=\boldsymbol{13^8}\) (or \(815730721\))
• \((36^3)^2=\boldsymbol{36^6}\) (or \(2176782336\))
• \(\frac{5^9}{5^3}=\boldsymbol{15625}\) (or \(5^6\))
• \(70^{-6}=\boldsymbol{\frac{1}{70^6}}\) (or \(\frac{1}{117649000000}\))