Question

simplify the expression. write your answer on the line using only positive exponents. (2 pts each)1. $\frac{15x^{3}y^{4}}{3x^{-5}y^{9}}$2. $(-5a^{4}b^{2})^{3}$3. $5^{-2} * 5^{5}$4. $(-4x^{2})^{3}$5. $(\frac{1}{3x})^{-4}$6. $(3x^{2})(4x^{4})$rewrite the expression using rational exponents or radicals. (1 pt each)7. $-sqrt5{22}$8. $(-4)^{\frac{3}{4}}$evaluate the expression. (1 pt each)9. $-sqrt3{216}$10. $(64)^{\frac{5}{3}}$11. simplify $(4.3 \times 10^{-3})(1.5 \times 10^{8})$ write your answer in both standard and scientific notation.scientific: _____standard: ___12. simplify $\frac{6.3 \times 10^{3}}{2 \times 10^{6}}$ write your answer in both standard and scientific notation.scientific: ___standard: _____

Explanation:

Step1: Simplify coefficients and $x$-terms

$\frac{15}{3} \cdot x^{3 - (-5)} = 5x^{8}$

Step2: Simplify $y$-terms, make exponent positive

$y^{4 - 9} = y^{-5} = \frac{1}{y^5}$

Step3: Combine results

$5x^{8} \cdot \frac{1}{y^5} = \frac{5x^8}{y^5}$

Step4: Apply power of a product rule

$(-5)^3 \cdot (a^4)^3 \cdot (b^2)^3$

Step5: Calculate each component

$-125 \cdot a^{12} \cdot b^{6} = -125a^{12}b^{6}$

Step6: Apply product of powers rule

$5^{-2 + 5} = 5^{3}$

Step7: Evaluate the power

$5^3 = 125$

Step8: Apply power of a product rule

$(-4)^3 \cdot (x^2)^3$

Step9: Calculate each component

$-64 \cdot x^{6} = -64x^6$

Step10: Rewrite as positive exponent power

$(3x)^{-4} = \frac{1}{(3x)^4}$

Step11: Expand denominator

$\frac{1}{3^4x^4} = \frac{1}{81x^4}$

Step12: Multiply coefficients, add $x$-exponents

$3 \cdot 4 \cdot x^{2 + 4} = 12x^6$

Step13: Rewrite radical as rational exponent

$-22^{\frac{1}{5}}$

Step14: Rewrite rational exponent as radical

$\sqrt[4]{(-4)^3}$

Step15: Evaluate cube root first

$-\sqrt[3]{216} = -6$

Step16: Rewrite power, evaluate stepwise

$(64^{\frac{1}{3}})^5 = 4^5 = 1024$

Step17: Multiply coefficients, add exponents

$4.3 \cdot 1.5 \times 10^{-3 + 8} = 6.45 \times 10^{5}$

Step18: Convert to standard notation

$6.45 \times 10^{5} = 645000$

Step19: Divide coefficients, subtract exponents

$\frac{6.3}{2} \times 10^{3 - 6} = 3.15 \times 10^{-3}$

Step20: Convert to standard notation

$3.15 \times 10^{-3} = 0.00315$

Answer:

1. $\frac{5x^8}{y^5}$
2. $-125a^{12}b^{6}$
3. $125$
4. $-64x^6$
5. $\frac{1}{81x^4}$
6. $12x^6$
7. $-22^{\frac{1}{5}}$
8. $\sqrt[4]{(-4)^3}$
9. $-6$
10. $1024$
11. Scientific: $6.45 \times 10^{5}$; Standard: $645000$
12. Scientific: $3.15 \times 10^{-3}$; Standard: $0.00315$