Question

Write your answer as a power or as a product of powers. $$ (-x)^{4}(-x)^{3}(-x)^{2} $$

Short answer

The simplification of the expression \((-x)^4(-x)^3(-x)^2\) is \(-x^9\).

Step-by-step solution

Apply the properties of exponents

When two terms with the same base are multiplied, the exponent becomes the sum of the individual exponents. So, in case of \((-x)^4\), \((-x)^3\) and \((-x)^2\), add the exponents to get \((-x)^{4+3+2}\).

Calculate the sum

Now, add the exponents together. \(4 + 3 + 2 = 9\). So, \((-x)^{4+3+2}\) simplifies to \((-x)^9\)

Consider the negative sign

When the exponent is odd, a negative base raised to this exponent will result in a negative value. So, the sign of the final answer will be negative.

Key concepts

Power of a Negative Base

When dealing with the power of a negative base, it's important to understand how the sign of the base and the value of the exponent interact. A negative base raised to an even exponent results in a positive value, while a negative base raised to an odd exponent results in a negative value. This is because multiplying a negative by a negative gives a positive, and doing this an even number of times will always result in a positive. On the contrary, for an odd number of multiplications, you'll end up with a negative product.

For example, consider \( (-3)^2 \). Here, the base \( -3 \) is negative, and the exponent is 2, which is even. Thus, \( (-3)^2 = 9 \), a positive result. Contrast this with \( (-3)^3 \) where the exponent is 3, which is odd. So, \( (-3)^3 = -27 \) gives us a negative result.

Simplifying Expressions

Simplifying expressions in algebra involves rewriting them in a more concise and readable form without changing their value. This often includes combing like terms, reducing fractions, and applying exponent rules. For expressions involving exponents, simplification usually involves combining terms with the same base by adding their exponents, as seen in the original exercise.

When simplifying, pay close attention to the base and its associated sign. For instance, the expression \( (-x)^4(-x)^3(-x)^2 \) simplifies to \( (-x)^{4+3+2} \) by adding the exponents while keeping the base the same. This step reduces the complexity of the expression and prepares you for further evaluation.

Exponent Rules

Adding Exponents (Multiplication Rule)

When multiplying two powers with the same base, you add the exponents. This rule is demonstrated as \( a^m \times a^n = a^{m+n} \). Remember, this rule only applies when the base is identical.

Product to a Power

When a product is raised to a power, each factor is raised to that power, as seen in \( (ab)^n = a^n \times b^n \).

Power to a Power

When taking a power to another power, multiply the exponents: \( (a^m)^n = a^{mn} \).

Using these rules properly can significantly simplify complex expressions and allow for easier computation.

Negative Exponents

Negative exponents indicate division rather than multiplication. The rule states that \( a^{-n} \) is the reciprocal of \( a^n \), written as \( a^{-n} = \frac{1}{a^n} \). This means that instead of multiplying the base by itself 'n' times, you divide 1 by the base 'n' times.

For example, \( 2^{-3} \) simplifies to \( \frac{1}{2^3} \) or \( \frac{1}{8} \). Understanding this rule is crucial for simplifying expressions that involve negative exponents. It allows the expression to be rewritten in a more conventional non-negative exponent form and reveals the true magnitude of values that negative exponents represent.