Problem 54 - Algebra and Trigonometry

Question

Simplify the expression and eliminate any negative exponents \((\mathrm{s}) .\) Assume that all letters denote positive numbers. \(\left(x^{-5} y^{3} z^{10}\right)^{-3 / 5}\)

Step-by-Step Solution

Answer

\(\frac{x^3}{z^6 y^{\frac{9}{5}}}\)

Step 1: Apply Power to a Power Rule

According to the power to a power rule, when raising a power to another power, we multiply the exponents. Start by applying this rule to each part of the expression: \( \left(x^{-5} y^{3} z^{10}\right)^{-\frac{3}{5}} \) becomes: \( x^{-5 \times -\frac{3}{5}} y^{3 \times -\frac{3}{5}} z^{10 \times -\frac{3}{5}} \).

Step 2: Simplify the Exponents

Multiply the exponents: - For \(x\): \(-5 \times -\frac{3}{5} = 3\)- For \(y\): \(3 \times -\frac{3}{5} = -\frac{9}{5}\)- For \(z\): \(10 \times -\frac{3}{5} = -6\)

Step 3: Rewrite with Positive Exponents

To eliminate negative exponents, rewrite the expression using positive exponents by flipping the terms: \(x^3 \cdot z^{-6} \cdot y^{-rac{9}{5}}\) becomes \(\frac{x^3}{z^6 y^{\frac{9}{5}}}\).

Step 4: Present Simplified Expression

The final expression with positive exponents is: \(\frac{x^3}{z^6 y^{\frac{9}{5}}}\). This is the simplified form without any negative exponents.

Key Concepts

Negative Exponents

A negative exponent signals that the base, which the exponent is attached to, should be moved to the other side of the fraction line. This means that if the base with a negative exponent is in the numerator, it moves to the denominator, and vice versa. The base then appears with a positive exponent in its new position.

For instance, when you have an expression like \(x^{-5}\), this is equivalent to \(\frac{1}{x^5}\). Essentially, the negative sign in the exponent instructs us to "flip" the base to the other part of the fraction, converting the exponent to positive.

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

This flipping property is quite helpful when simplifying expressions, especially those involving multiple terms with different negative exponents.

Power to a Power Rule

The power to a power rule is a fundamental principle in exponentiation that simplifies expressions involving exponents. When raising a power to another power, you multiply the exponents together. This rule helps in breaking down complex expressions easily.

For example, if you have \((a^m)^n\), according to the power to a power rule, it is simplified to \(a^{m \times n}\). This rule reduces the number of steps needed to solve or simplify exponential expressions by allowing simple multiplication of the exponents.

• \((x^a)^b = x^{a \times b}\)
• \((x^{-a})^{-b} = x^{a \times b}\) because two negatives make a positive

In our original exercise, applying this rule to \(x^{-5}\), \(y^3\), and \(z^{10}\) when raised to \(-\frac{3}{5}\), helps simplify each term's exponent efficiently.

Exponentiation Rules

Exponentiation rules are the cornerstone of operations involving powers and exponents in mathematics. These rules allow you to simplify expressions more efficiently.

Some of the essential rules include:

• Multiplication of Bases with Same Exponents: \(a^m \times a^n = a^{m+n}\) - Here, you add the exponents while keeping the base the same.
• Division of Bases with Same Exponents: \(a^m \div a^n = a^{m-n}\) - When dividing, subtract the exponents.
• Zero Exponent Rule: \(a^0 = 1\) - Any base raised to the power of zero equals one.
• Negative Exponent Rule: See above.

By applying these rules, managing expressions such as the one we simplified becomes easier. It’s all about manipulating the base and exponents according to these guidelines, ensuring a streamlined approach to find the simplest form of expressions.