Question

Which real number satisfies \(\left(2^{x}\right)(4)=8^{3} ?\) F. 2 G. 3 H. 4 J. 4.5 K. 7

Short answer

Answer: K. 7

Step-by-step solution

Rewrite the equation with the same base

We can rewrite the equation \((2^x)(4) = 8^3\) by expressing 4 and 8 in terms of base 2: \(2^x * 2^2 = (2^3)^3\).

Apply exponent properties

Now, we can apply exponent properties to the equation. Using the product of powers property, we have \((2^x * 2^2) = (2^{x + 2})\). And using the power of a power property, we have \((2^3)^3 = 2^{(3 * 3)}\) which gives us \(2^{x + 2} = 2^9\).

Compare exponents

Since the bases (2) are the same, and the equation \(2^{x + 2} = 2^9\) is true, we can compare the exponents: \(x + 2 = 9\).

Solve for x

Now we can solve for x: \(x = 9 - 2 = 7\).

Select the correct answer

Comparing our result with the given choices:\begin{align*}F. 2 \\G. 3 \\H. 4 \\J. 4.5 \\K. 7\end{align*} We find that the correct answer is \(\boxed{\text{K. } 7}\).

Key concepts

ACT Math Preparation

Succeeding in the math section of the ACT requires a solid understanding of various mathematical concepts, one of which is exponents and powers. To prepare effectively for questions involving exponents, you should understand the rules that govern their operations and practice applying these rules in solving equations.

When studying for the ACT, focus on mastering the fundamental properties of exponents, such as the product of powers, quotient of powers, and power of a power. This will allow you to simplify and solve exponential equations efficiently. Additionally, practice with real ACT problems and solutions, such as the given exercise, enabling you to familiarize yourself with the format and difficulty level of the questions you will encounter on the actual test.

Regular drilling with similar types of exponential equations can build speed and accuracy. Utilize online resources, practice tests, and textbook exercises to enhance your problem-solving skills and increase your confidence in tackling this portion of the ACT.

Exponents and Powers

The concept of exponents and powers is a fundamental building block in mathematics. An exponent represents the number of times a base is multiplied by itself. For example, in the expression \(2^3\), 2 is the base, and 3 is the exponent, indicating that 2 should be multiplied by itself 3 times (2 * 2 * 2).

Properties of Exponents

To simplify expressions and solve equations involving exponents, we rely on several exponent properties, such as:

• Product of Powers: \(a^m \cdot a^n = a^{m+n}\)
• Quotient of Powers: \(\frac{a^m}{a^n} = a^{m-n}\)
• Power of a Power: \(\left(a^m\right)^n = a^{m \cdot n}\)
• Power of a Product: \(\left(ab\right)^n = a^n \cdot b^n\)
• Negative Exponents: \(a^{-n} = \frac{1}{a^n}\) if \a e 0\

Understanding these properties allows students to manipulate and solve exponential expressions with confidence, particularly in Algebra and on standardized tests such as the ACT.

Solving Exponential Equations

An exponential equation is an equation in which a variable appears in the exponent. Solving exponential equations often involves making the bases on both sides of the equation the same, so you can then equate and solve for the exponent. As shown in the step-by-step solution, to solve \(2^x\cdot 4 = 8^3\), we need to express all terms with the same base and use the properties of exponents.

Here’s a simplified strategy to solve such equations:

1. Rewrite the equation with a common base if possible.
2. Apply the properties of exponents to simplify the equation.
3. When the bases are the same, set the exponents equal to each other and solve for the variable.
4. Check that your solution corresponds with one of the given options, if applicable.

By following these steps, which include the exercise improvement advice, you learn to systematically approach and solve exponential equations, a skill that is highly beneficial not only for standardized tests but also for higher-level math courses.