Podcast
Questions and Answers
What is the simplified form of $(a^2 b^3) imes (b^2)$?
What is the simplified form of $(a^2 b^3) imes (b^2)$?
- a^1 b^3
- a^2 b^6
- a^2 b^5 (correct)
- a^2 b^1
What is the result of simplifying $(p^4) imes (q^3)$?
What is the result of simplifying $(p^4) imes (q^3)$?
- p^4 q^3 (correct)
- p^7 q^3
- p^4 + q^3
- p^4 q^4
When simplifying $(f^8)/(f^3)$, what is the result?
When simplifying $(f^8)/(f^3)$, what is the result?
- f^5 (correct)
- f^11
- f^3
- f^8
What is the simplified version of $(3w^9 q^2) imes (2w^3)$?
What is the simplified version of $(3w^9 q^2) imes (2w^3)$?
What happens to the expression $(g^3 h^2)^{2}$ when simplified?
What happens to the expression $(g^3 h^2)^{2}$ when simplified?
What is the simplified form of $(2a) imes (3a^2)$?
What is the simplified form of $(2a) imes (3a^2)$?
What is the result of simplifying the expression $(e^2)^{3}$?
What is the result of simplifying the expression $(e^2)^{3}$?
When simplifying $(f^4) imes (f^8) imes (f^2)$, what is the result?
When simplifying $(f^4) imes (f^8) imes (f^2)$, what is the result?
What is the simplified form of $(−1)^6$?
What is the simplified form of $(−1)^6$?
Which statement correctly explains why Jo's assertion that $−x^2$ is equivalent to $(−x)^2$ is incorrect?
Which statement correctly explains why Jo's assertion that $−x^2$ is equivalent to $(−x)^2$ is incorrect?
What is the correct calculation for $54$?
What is the correct calculation for $54$?
When simplifying $8^x = 32$, what base do both sides share after simplification?
When simplifying $8^x = 32$, what base do both sides share after simplification?
Which of the following is true regarding the reason all three equations ($8^x = 32$, $27^x = 243$, $1000^x = 100000$) have the same solution?
Which of the following is true regarding the reason all three equations ($8^x = 32$, $27^x = 243$, $1000^x = 100000$) have the same solution?
What is the simplified form of $r^6 \div r^4$?
What is the simplified form of $r^6 \div r^4$?
What is the simplified form of $(a^3 \div a^2)$?
What is the simplified form of $(a^3 \div a^2)$?
What is the result of $(m^8 \div m^3)$?
What is the result of $(m^8 \div m^3)$?
If $(p^9 \div p^6)$ is simplified, what is the result?
If $(p^9 \div p^6)$ is simplified, what is the result?
If you rewrite the expression $47 \times 47 \times 47 \times 47 \times 47$ using an exponent, what is the exponent?
If you rewrite the expression $47 \times 47 \times 47 \times 47 \times 47$ using an exponent, what is the exponent?
What is the result of $(n^5 \div n^6)$?
What is the result of $(n^5 \div n^6)$?
What is the simplified form of $(b^4 \div b^6)$?
What is the simplified form of $(b^4 \div b^6)$?
What is the outcome of $(f^7 \div f^2)$?
What is the outcome of $(f^7 \div f^2)$?
What is the simplified form of $3^2 \times 3^4$?
What is the simplified form of $3^2 \times 3^4$?
What is the simplified result of $(g^7) \times (g^9)$?
What is the simplified result of $(g^7) \times (g^9)$?
What is the result of $(2d^7) \times (3d^2)$?
What is the result of $(2d^7) \times (3d^2)$?
What is the simplified form of $(10r^2) \times (2r^3)$?
What is the simplified form of $(10r^2) \times (2r^3)$?
What does $(p^7) \div (p^2)$ simplify to?
What does $(p^7) \div (p^2)$ simplify to?
What is the simplified result of $(t^7) \times (t^3)$?
What is the simplified result of $(t^7) \times (t^3)$?
Which of the following is the result of $(3a^2) \times (2a^6)$?
Which of the following is the result of $(3a^2) \times (2a^6)$?
What is the simplified form of $(w^5) \times (p^7)$?
What is the simplified form of $(w^5) \times (p^7)$?
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Study Notes
Simplifying Algebraic Products and Quotients
- When multiplying terms with the same base, add the exponents.
- When dividing terms with the same base, subtract the exponents.
- When raising a term to a power, multiply the exponents.
Simplifying Algebraic Terms with Powers
- (e^2)^3 = e^(2 * 3) = e^6 (Example of simplifying a term raised to a power)
- (a^2 b^3)^4 = a^(2 * 4) b^(3 * 4) = a^8 b^12 (Example of simplifying a product of terms raised to a power)
- (3w^9 q^2)^4 = 3^4 w^(9 * 4) q^(2 * 4) = 81w^36 q^8 (Example of simplifying a term with a coefficient raised to a power)
Understanding Index Laws
- Index laws allow for simplification of expressions with exponents.
- Multiplication of terms with the same base: a^m x a^n = a^(m + n)
- Division of terms with the same base: a^m ÷ a^n = a^(m - n)
- Power of a power: (a^m)^n = a^(m x n)
Negative Exponents
- When raising -1 to a positive power, the result alternates between -1 and 1.
- (-1)^10 = 1 (Example of raising -1 to an even power)
- (-1)^7 = -1 (Example of raising -1 to an odd power)
Solving Equations with Exponents
- 8^x = 32 can be rewritten as 2^(3x) = 2^5. Therefore, 3x = 5 and x = 5/3.
- All equations in the form a^x = a^y have the same solution, x = y. This is because the base is the same.
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