Simplifying Expressions Quiz

What is the power rule for raising a monomial to a power?

Answer explanation

The power rule states that when raising a monomial to a power, you multiply the exponents. Thus, (x^a)^b = x^{ab} is correct, as it reflects this multiplication of exponents.

Answer explanation

To simplify (x^2)^3, use the power of a power rule: multiply the exponents. Thus, (x^2)^3 = x^(2*3) = x^6. Therefore, the correct answer is x^6.

Answer explanation

To simplify (2x^2y)^5, apply the power to each factor: (2^5)(x^{2*5})(y^5) = 32x^{10}y^5. Thus, the correct answer is 32x^{10}y^5.

Answer explanation

To simplify (-8c^4d^9)^2, square the coefficient and each variable: (-8)^2 = 64, (c^4)^2 = c^8, and (d^9)^2 = d^{18}. Thus, the expression simplifies to 64c^8d^{18}, which is the correct answer.

Answer explanation

To simplify (-5m^6n)^4, apply the power to each factor: (-5)^4 = 625, (m^6)^4 = m^{24}, and (n)^4 = n^4. Thus, the expression simplifies to 625m^{24}n^4, which matches the correct choice.

Answer explanation

To simplify \left(\frac{1}{2}a^3b^6c^5\right)^3, apply the power of a product rule: \left(\frac{1}{2}\right)^3 a^{3\cdot3} b^{6\cdot3} c^{5\cdot3} = \frac{1}{8} a^9 b^{18} c^{15}. Thus, the correct answer is \frac{1}{8}a^9b^{18}c^{15}.

Answer explanation

To simplify (x^3y^3) \cdot x^2, use the property of exponents: x^a \cdot x^b = x^{a+b}. Here, x^3 \cdot x^2 = x^{3+2} = x^5. The y term remains y^3. Thus, the simplified expression is x^5y^3, which is the correct answer.

Answer explanation

To simplify, first calculate each part: (-ab^7)^2 = a^2b^{14} and (12a^3b)^2 = 144a^6b^2. Now multiply: a^2b^{14} * 144a^6b^2 = 144a^{8}b^{16}. Thus, the correct answer is 144a^8b^{16}.

Answer explanation

To simplify, first square \( \frac{1}{3}abc \) to get \( \frac{1}{9}a^2b^2c^2 \). Then multiply: \( (6a^7b^3) \cdot (\frac{1}{9}a^2b^2c^2) = \frac{6}{9}a^{7+2}b^{3+2}c^2 = \frac{2}{9}a^9b^5c^2 \). Thus, the correct answer is \( \frac{2}{9}a^9b^5c^2 \).

Answer explanation

To simplify (2a^3) + (a^4)(3a^2), first calculate (a^4)(3a^2) = 3a^{4+2} = 3a^6. Now combine like terms: 2a^3 + 3a^6 = 3a^6 + 2a^3. Thus, the correct answer is 3a^6 + 2a^3.