Problem 42 - Beginning and Intermediate Algebra

Question

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

Step-by-Step Solution

Answer

The simplified expression for \(\left(-3 x^{2}\right)^{4}\) is \(81x^8\).

Step 1: Raise Each Term Inside the Parenthesis to the Power

We can distribute the power of 4 to both terms inside the parenthesis: -3 and \(x^2\). This gives us: \((-3)^4 \cdot \left(x^{2}\right)^{4}\)

Step 2: Apply the Exponent Rule for Raising a Power to a Power

Next, we apply the rule for raising a power to a power: \(a^{m^e}=a^{m \cdot{e}}\). This will apply to the \((x^2)^4\) term in our expression. By applying this rule, we get: \((-3)^4 \cdot x^{(2 \cdot 4)}\)

Step 3: Evaluate the Exponents

Now we can evaluate each exponent. \((-3)^4\) is equal to \(3^4\), which is \(81\). \(x^{(2 \cdot 4)}\) is equal to \(x^8\). So, our simplified expression is: \(81 \cdot{x^8}\)

Step 4: Write the Final Answer

The given expression, \(\left(-3 x^{2}\right)^{4}\), simplifies to: \(81x^8\)

Key Concepts

Exponent Rules

Learning and applying exponent rules is a crucial skill in algebra. These rules help us simplify expressions with powers in a systematic way.
One of the main rules applied in our example is that when multiplying identical bases, you add the exponents. For example, if you have \( a^m \) and \( a^n \), multiplying these terms gives you \( a^{m+n} \). However, when raising a power to a power, the rules are different.
Another helpful rule is that any non-zero number raised to the power of zero is 1, and this can help simplify more complex expressions in other examples. These rules ensure consistency and accuracy when dealing with powers.

Power of a Power Rule

The power of a power rule is a unique exponentiation rule that truly shines when you're simplifying expressions with exponents.
This rule states that when you have a power raised to another power, you multiply the exponents together. In mathematical terms, \((a^m)^n = a^{m \cdot n}\).
In our exercise, this rule helped simplify \((x^2)^4\) to \(x^{2 \cdot 4}\), which equals \(x^8\). The rule is especially useful for keeping track of exponents in more complicated expressions. It makes it easier to manage and simplify them effectively.

Distributive Property in Exponents

The distributive property of exponents allows you to apply an exponent to each term within a parenthesis separately.
In math, you might often see it applied like this: \((ab)^n = a^n \cdot b^n\). In our example, this property allowed us to distribute the exponent of 4 to both \(-3\) and \(x^2\).
So, \((-3x^2)^4\) became \((-3)^4 \cdot (x^2)^4\). Applying the distributive property in exponents makes it straightforward to handle expressions with multiple terms efficiently. This technique makes simplifying expressions much more manageable.