Question

simplify each expression using the exponent rules with positive exponents only.

1. $(3a^{3}b^{4})^{3}$
2. $(5m^{4}n^{-5})(2m^{-1}n^{6})$
3. $6^{-3}6^{7}$
4. $\frac{5(x^{5})}{12x^{-3}}$
5. $\frac{(4x^{3})(3x^{6})^{2}}{2x^{-2}}$

simplify each expression completely.

1. $7\sqrt{45x^{3}}$
2. $-2\sqrt{80x^{5}y^{6}}$
3. $sqrt{192km^{2}}$

Explanation:

Step1: Apply power of a product rule

$(3a^3b^4)^3 = 3^3 \cdot (a^3)^3 \cdot (b^4)^3$

Step2: Calculate exponents and constants

$3^3=27$, $(a^3)^3=a^{3 \times 3}=a^9$, $(b^4)^3=b^{4 \times 3}=b^{12}$
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Step1: Multiply coefficients, combine like bases

$(5m^4n^{-5})(2m^{-1}n^6) = (5 \times 2) \cdot m^{4+(-1)} \cdot n^{-5+6}$

Step2: Simplify each part

$5 \times 2=10$, $m^{4-1}=m^3$, $n^{-5+6}=n^1=n$
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Step1: Apply product of powers rule

$6^{-3}6^7 = 6^{-3+7}$

Step2: Simplify the exponent

$6^{4}=1296$
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Step1: Rewrite negative exponent as positive

$\frac{5(x^5)}{12x^{-3}} = \frac{5x^5 \cdot x^3}{12}$

Step2: Combine like bases

$\frac{5x^{5+3}}{12} = \frac{5x^8}{12}$
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Step1: Expand the squared term

$\frac{(4x^3)(3x^6)^2}{2x^{-2}} = \frac{(4x^3)(9x^{12})}{2x^{-2}}$

Step2: Multiply numerator terms

$\frac{4 \times 9 \cdot x^{3+12}}{2x^{-2}} = \frac{36x^{15}}{2x^{-2}}$

Step3: Divide coefficients, combine exponents

$\frac{36}{2} \cdot x^{15-(-2)} = 18x^{17}$
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Step1: Factor radicand into perfect squares

$7\sqrt{45x^3} = 7\sqrt{9 \times 5 \times x^2 \times x}$

Step2: Simplify the square root

$7 \times 3x\sqrt{5x} = 21x\sqrt{5x}$
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Step1: Factor radicand into perfect squares

$-2\sqrt{80x^5y^6} = -2\sqrt{16 \times 5 \times x^4 \times x \times y^6}$

Step2: Simplify the square root

$-2 \times 4x^2y^3\sqrt{5x} = -8x^2y^3\sqrt{5x}$
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Step1: Factor radicand into perfect squares

$\sqrt{192km^2} = \sqrt{64 \times 3 \times k \times m^2}$

Step2: Simplify the square root

$8|m|\sqrt{3k}$

Answer:

1. $27a^9b^{12}$
2. $10m^3n$
3. $1296$
4. $\frac{5x^8}{12}$
5. $18x^{17}$
6. $21x\sqrt{5x}$
7. $-8x^2y^3\sqrt{5x}$
8. $8|m|\sqrt{3k}$