Problem 65 - Elementary Algebra

Question

\(\left(-x^{4}\right)^{2}\left(-x^{2}\right)^{3}\)

Step-by-Step Solution

Answer

-x^{14}

Step 1: Simplify each term separately

First, simplify \(\text{(-x}^4\text{)}^2\). This is equal to \((-x^4)^2 = (-x^4)(-x^4) = x^8\). The negative sign disappears because a negative times a negative is a positive.

Step 2: Simplify the second term

Next, simplify \((-x^2)^3\). This is equal to \((-x^2)^3 = (-x^2)(-x^2)(-x^2) = -x^6\). The result of an odd power of a negative number is negative, so it remains as \(-x^6\).

Step 3: Multiply the simplified terms

Now multiply the simplified terms together: \(x^8 \times -x^6\). When multiplying powers with the same base, add the exponents: \(x^8 \times -x^6 = -x^{14}\).

Step 4: Final Step: Combine the results

Thus, the simplified form of \(\text{(-x}^4\text{)}^2 (\text{-x}^2\text{)}^3\) is \(-x^{14}\).

Key Concepts

negative exponents

Negative exponents can seem tricky at first, but they follow a simple rule: they indicate the reciprocal of the base raised to the opposite positive exponent. For example, if you have an expression like \((a^{-n})\), this can be rewritten as \(\frac{1}{a^n}\)\.

This concept is useful when simplifying expressions because it can often make an expression easier to work with.
Remember these key points about negative exponents:

• \((a^{-n}) = \frac{1}{a^n}\)
• \((\frac{1}{a^{-n}} = a^n\)

Negative exponents do not make the base negative; they only change the position of the base in a fraction. Practice moving terms between the numerator and the denominator to become comfortable with this rule.

multiplying powers

When multiplying powers with the same base, you add the exponents. This rule simplifies expressions quickly. Let's say you have the expression \(x^3 \times x^2\), you would add the exponents 3 and 2 to get \((x^{3 + 2} = x^5)\).

Another example is with an expression having a negative exponent like \(x^4 \times x^{-2}\). Here, you would add the exponents 4 and -2 to get \((x^{4 + (-2)} = x^2)\).

• Same bases are the key point
• Adding exponents only applies to multiplication

Be wary of different bases or additional operations within the same expression, as these change how you apply the rules.

simplifying expressions

Simplifying expressions helps to make them easier to understand and work with. This is achieved by combining like terms and reducing the expression to its simplest form.

• First, perform any operations inside parentheses.
• Apply the rules of exponents: multiply terms with the same base by adding exponents.

In our example, \( (-x^4)^2(-x^2)^3 \), we first simplify each component separately. Breaking it down step-by-step:

1. Simplify \( (-x^4)^2 = (x^8) \).
2. Simplify \((-x^2)^3 = (-x^6)\).

Finally, the simplified components are multiplied together by adding the exponents: \(x^8 \times -x^6 = -x^{14} \). Simplifying expressions like this breaks complex problems into understandable parts.