Problem 56 Simplify if possible:\(\left(a^{... [FREE SOLUTION]

Chapter 1: Problem 56

Simplify if possible:\(\left(a^{2}\right)^{-3}\)

Short Answer

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\(\frac{1}{a^6}\)

Step by step solution

Apply the Power Rule of Exponents

When raising an exponent to another power, multiply the exponents. So, \[\left(a^{2}\right)^{-3} = a^{2 \times -3}\] which simplifies to \[a^{-6}\].

Convert Negative Exponent to Positive

Rewrite the expression with a positive exponent. Negative exponents mean "take the reciprocal", so: \[a^{-6} = \frac{1}{a^6}\].

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Power Rule of Exponents

When you see an expression like \((a^2)^{-3}\), you're looking at a case for the Power Rule of Exponents. This rule is pretty straightforward: when you're raising a power to another power, you simply multiply the exponents together. Think of it as combining the repeated operations. For example, with \((a^2)^{-3}\), you're multiplying 2 (the exponent from \(a^2\)) by -3 (the exponent on the outside), resulting in \(a^{2 \times -3}\), which simplifies to \(a^{-6}\). This rule helps simplify expressions and makes complex calculations manageable.

Negative Exponents

Negative exponents might look a bit tricky at first, but they're not too hard to handle once you understand the concept. Essentially, a negative exponent indicates that you need to take the reciprocal of the base raised to the corresponding positive exponent. For example, the expression \(a^{-6}\) is the same as \(\frac{1}{a^6}\). These exponents tell us that instead of multiplying, we're essentially dividing. It's like flipping the base to the denominator, getting rid of the negative on the exponent. This concept is quite powerful in mathematics, allowing simplification and reformatting of expressions that might initially seem difficult to work with.

Simplifying Expressions

Simplifying expressions is all about reducing them to their simplest form while maintaining the equality in value. This involves using various rules, like the power rule and handling negative exponents, to make expressions clearer or easier to compute. For the given exercise, simplifying \((a^2)^{-3}\) involves applying these rules systematically:

First, apply the power rule to get \(a^{-6}\), simplifying the expression from being nested.
Next, convert the negative exponent into a positive one using reciprocals, resulting in \(\frac{1}{a^6}\).

By following these steps, we take a complex-looking expression and turn it into something more straightforward. Practicing these skills can help make complex mathematics less daunting and more intuitive.

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91影视

Simplify if possible:\(\left(a^{2}\right)^{-3}\)

Short Answer

Step by step solution

Apply the Power Rule of Exponents

Convert Negative Exponent to Positive

Key Concepts

Power Rule of Exponents

Negative Exponents

Simplifying Expressions

One App. One Place for Learning.

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