In Exercises 35–52, write each expression with positive exponents only. Then simplify, if possible.a⁻⁴b⁷/c⁻³

Verified step by step guidance

1. Start by addressing the negative exponents in the expression \( \frac{a^{-4}b^7}{c^{-3}} \). Recall that a negative exponent indicates the reciprocal of the base raised to the positive of that exponent.
2. Rewrite \( a^{-4} \) as \( \frac{1}{a^4} \) and \( c^{-3} \) as \( \frac{1}{c^3} \).
3. Substitute these into the expression: \( \frac{1}{a^4} \cdot b^7 \cdot c^3 \).
4. Combine the terms into a single fraction: \( \frac{b^7c^3}{a^4} \).
5. The expression is now written with positive exponents only and is simplified as \( \frac{b^7c^3}{a^4} \).

Key Concepts

Negative Exponents

Negative exponents indicate the reciprocal of the base raised to the absolute value of the exponent. For example, a⁻⁴ can be rewritten as 1/a⁴. This concept is essential for transforming expressions with negative exponents into forms that only contain positive exponents.

Simplifying Expressions

Simplifying expressions involves reducing them to their simplest form by combining like terms and applying the laws of exponents. This includes operations such as multiplying and dividing bases with exponents, which can lead to further simplification of the expression.

Laws of Exponents

The laws of exponents are rules that govern how to manipulate expressions involving exponents. Key laws include the product of powers (a^m * a^n = a^(m+n)), the quotient of powers (a^m / a^n = a^(m-n)), and the power of a power ( (a^m)^n = a^(m*n)). Understanding these laws is crucial for rewriting and simplifying expressions correctly.