Question

Remove the brackets from \(\left(7 x^{2}\right)^{-3} .\)

Short answer

Question: Remove the brackets from the expression \(\left(7 x^{2}\right)^{-3}\) and simplify. Answer: \(\frac{1}{343}x^{-6}\)

Step-by-step solution

Distribute the exponent

Distribute the exponent \(-3\) to both the coefficient \(7\) and the variable \(x^{2}\) inside the brackets. \((7x^{2})^{-3} \Rightarrow 7^{-3}(x^{2})^{-3}.\)

Apply exponent rules for the variable

Apply the exponent rules for the variable \(x^{2}\). Remember that when taking the power of a power, we multiply the exponents. So, \((x^{2})^{-3} \Rightarrow x^{(2 \times -3)} \Rightarrow x^{-6}\).

Simplify the coefficient

Now, simplify the coefficient \(7^{-3}\) which means \(\frac{1}{7^3}\) or \(\frac{1}{343}\).

Write the final result

Combine the simplified coefficient and variable to obtain the final simplified expression. The simplified expression is \(\frac{1}{343}x^{-6}\).

Key concepts

Exponent Rules

Exponent rules are foundational principles that govern how we handle expressions involving exponents. These rules help us simplify and manipulate expressions more easily. Here are some key exponent rules you should know:

• Product of Powers Rule: When multiplying two powers with the same base, we add the exponents: \(a^m \cdot a^n = a^{m+n}\).
• Quotient of Powers Rule: When dividing two powers with the same base, we subtract the exponents: \(\frac{a^m}{a^n} = a^{m-n}\).
• Power of a Product Rule: When raising a product to an exponent, you apply the exponent to each factor in the product: \((ab)^m = a^m \cdot b^m\).
• Power of a Power Rule: When raising a power to another power, multiply the exponents: \((a^m)^n = a^{m\cdot n}\).

Simplifying expressions using these rules can make complex calculations much more manageable. It's like following a recipe to bake a cake - when you stick to the correct steps, you end up with a successful product.

Negative Exponents

Negative exponents are a special type of exponent that indicate the reciprocal of the base raised to the corresponding positive exponent. When you see a negative exponent, it tells you to take the reciprocal:

• For any non-zero number \(a\) and a positive integer \(n\), \(a^{-n} = \frac{1}{a^n}\).

This means that instead of multiplying the base by itself, you are dividing 1 by the base multiplied by itself the given number of times. For example, \(2^{-3} = \frac{1}{2^3} = \frac{1}{8}\).

Understanding negative exponents helps in simplifying expressions, especially when combined with other exponent rules. It allows you to express powers in a different form that might be easier to work with in certain mathematical operations. So the next time you encounter them, remember they are just demanding you to take the reciprocal of a positive power!

Power of a Power Rule

The power of a power rule is crucial when you deal with nested exponents — that is, a power raised to another power. The rule simplifies the expression by allowing you to multiply the exponents together.For example, if you have \((x^2)^{-3}\), using the power of a power rule means calculating \(x^{2\times(-3)} = x^{-6}\).This rule allows you to neatly condense expressions by converting double exponentials into a single operation.

This concept is helpful when working with more complex algebraic expressions, as it plays a key role in simplifying the terms. Keep in mind that when facing multiple nested powers, consistently applying this rule ensures clarity and accuracy. This way, the expressions are easier to manage and understand.