Question

Which of the following equations is INCORRECT? A. \(A^{3} \times B^{3}=(A B)^{3}\) B. \(A^{5} \div A^{7}=A^{-2}\) C. \(\left(A^{0.5}\right)^{4}+A^{2}=2 A^{2}\) D. \(\left(A^{3}\right)^{2}=A^{9}\)

Short answer

The incorrect equation is D.

Step-by-step solution

Examine Equation A

Analyze the equation \(A^{3} \times B^{3}=(A B)^{3}\). Use the property of exponents that states \(x^{a} \times y^{a} = (xy)^{a}\). Both sides are equal, thus it is correct.

Examine Equation B

Analyze the equation \(A^{5} \rightarrow div \; A^{7}=A^{-2}\). Use the property of exponents that states \(A^{m} \rightarrow div \; A^{n}=A^{m-n}\). So, \(A^{5} \rightarrow div \; A^{7}=A^{5-7}=A^{-2}\). This equation is correct.

Examine Equation C

Analyze the equation \(\rightarrow left(A^{0.5}\right)^{4}+A^{2}=2A^{2}\). Simplify \(\rightarrow left(A^{0.5}\right)^{4}\) to get \(A^{2} + A^{2} = 2A^{2}\). This equation is correct.

Examine Equation D

Analyze the equation \(\rightarrow left(A^{3}\right)^{2}=A^{9}\). Use the power property \((x^a)^b = x^{ab}\). So, \(\rightarrow left(A^{3}\right)^2=A^{3 \times 2}=A^{6}\), not \(A^{9}\). This equation is incorrect.

Key concepts

Properties of Exponents

In understanding exponents, properties play a crucial role. Here, we'll cover the most common properties which help simplify exponential expressions.

• Product of Powers: When multiplying two exponents with the same base, you add the exponents: \(x^a \times x^b = x^{a+b}\).
• Quotient of Powers: When dividing two exponents with the same base, subtract the exponents: \(x^a \rightarrow div \, x^b = x^{a-b}\).
• Power of a Power: When raising an exponent to another exponent, multiply the exponents: \((x^a)^b = x^{a \times b}\).
• Power of a Product: When raising a product to an exponent, apply the exponent to each factor: \((xy)^a = x^a \times y^a\).
• Zero Exponent Rule: Any nonzero number raised to the power of zero is 1: \(x^0 = 1\).

These properties are fundamental for simplifying and solving exponential equations.

Exponent Simplification

Exponent simplification involves using the properties of exponents to break down complex expressions.

Let's take Equation A from the exercise: \(A^3 \times B^3 = (AB)^3\).
Using the property of powers of a product, we know this is true. We apply the exponent to each factor inside the parentheses: \((AB)^3 = A^3 \times B^3 \).

Another example is Equation B: \(A^5 \rightarrow div \, A^7 = A^{-2}\).
We use the quotient of powers' property here. When we divide exponents with the same base, we subtract the exponents: \(A^{5-7} = A^{-2}\).
Simplifying exponents can make complex expressions more manageable and help in mathematical problem-solving.

Mathematical Analysis

Mathematical analysis involves carefully examining and manipulating expressions to determine their correctness or to simplify them.

Consider Equation D: \((A^3)^2 = A^9\).
By analyzing, we use the power of a power property: \((x^a)^b = x^{a \times b}\).
So, \((A^3)^2 = A^{3 \times 2} = A^6\), not \(A^9\).

For Equation C, \((A^{0.5})^4 + A^2 = 2A^2\):
Simplifying \((A^{0.5})^4\) gives \(A^{0.5 \times 4} = A^2\). Therefore the equation becomes \(A^2 + A^2 = 2A^2\), which is correct.

Analysis of equations using exponents involves checks and understanding of base rules to determine correctness and simplify expressions. This is essential for solving MCAT Physics and Math problems effectively.