Question

If \(\left(i^{413}\right)\left(i^x\right)=1\), then what is one possible value of \(x\) ? A) 0 B) 1 C) 2 D) 3

Short answer

D) 3

Step-by-step solution

Simplify \(i^{413}\)

To simplify \(i^{413}\), we need to find the remainder when the exponent (413) is divided by 4: Remainder of \(\frac{413}{4} = 1\). This means that \(i^{413} = i^1 = i\).

Substitute the simplified \(i^{413}\) into the equation

Now that we know that \(i^{413} = i\), substitute it back into the original equation: \((i)(i^x) = 1\)

Solve for x using the properties of exponents

\((i)(i^x) = 1\) can also be written as: \(i^{1+x} = 1\) Now, let's think about the powers of i: - \(i^0 = 1\) - \(i^1 = i\) - \(i^2 = -1\) - \(i^3 = -i\) - \(i^4 = 1\) (cycle begins again) We need to find a value of x such that \(i^{1+x} = 1\). From the list above, we can see that this happens when x = 3. Therefore, the possible value of x is: x = 3 (which corresponds to option D).

Key concepts

Complex Numbers

Complex numbers are a fundamental concept in mathematics, particularly in higher algebra and calculus. In SAT math practice, understanding complex numbers can often be pivotal to solving complex equations and problems effectively. A complex number is formed with two parts: the real part and the imaginary part. The imaginary unit is represented by the letter 'i', which is defined as the square root of -1. Therefore, we have that \( i^2 = -1 \).

When dealing with complex numbers, particularly powers of \(i\), you'll find that they follow a specific cyclic pattern: \(i^0 = 1\), \(i^1 = i\), \(i^2 = -1\), \(i^3 = -i\), and \(i^4 = 1\), after which the cycle repeats. This cycle is crucial for solving problems where the powers of \(i\) are involved. It allows us to simplify any power of \(i\) by finding its remainder when divided by 4, and then using this cycle to find the equivalent value.

Exponent Rules

Grasping exponent rules is essential for numerous algebra and pre-calculus topics, including SAT math practice problems. The rules of exponents dictate how to handle the multiplication, division, and raising of powers to powers when working with exponents. Key exponent rules include:

• Product of Powers: To multiply powers with the same base, you add the exponents; for example, \( a^m \times a^n = a^{m+n} \).
• Quotient of Powers: To divide powers with the same base, you subtract the exponents; for example, \( \frac{a^m}{a^n} = a^{m-n} \).
• Power of a Power: To raise a power to another power, you multiply the exponents; for example, \( (a^m)^n = a^{mn} \).

Knowing these rules allows you to simplify expressions with exponents quickly, and it exactly serves to easily solve for \(x\) when \(i^{1+x} = 1\), as in our original SAT mathematical problem.

SAT Problem Solving

SAT problem solving involves a combination of knowledge, strategic thinking, and time management. Key to success is understanding the underlying concepts, as applied in various contexts. For example, in our original exercise, problem-solving requires knowledge of complex numbers and exponent rules.

To approach SAT problem-solving questions effectively, practice these strategies:

• Analyze the given problem and identify what mathematical concepts it involves.
• Become familiar with the format of the questions to reduce surprise and stress during the test.
• Practice with a variety of problems to build a strong foundation across all relevant math concepts.
• Look for patterns or cycles, as often found in problems involving complex numbers, to simplify the problem.

With these strategies and a clear understanding of complex numbers and exponent rules, tackling SAT math practice problems can become more manageable, increasing the chances of selecting the correct answer and achieving a desirable score.