Problem 131 - Algebra A Combined Function

Question

Simplify each expression. Assume that variables represent positive integers. $$ \left(x^{-3 s}\right)^{3} $$

Step-by-Step Solution

Answer

The simplified expression is \(\frac{1}{x^{9s}}\).

Step 1: Implement the Power of a Power Rule

The expression \((x^{-3s})^3\) involves raising a power to another power. According to the Power of a Power Rule, \((a^m)^n = a^{m imes n}\). Therefore, we multiply the exponents: \[(x^{-3s})^3 = x^{-3s \times 3}\].

Step 2: Simplify the Exponents

Next, multiply the exponents: \[-3s \times 3 = -9s\]. Thus, the expression simplifies to: \[x^{-9s}\].

Step 3: Apply the Negative Exponent Rule

An expression with a negative exponent means that the base is on the wrong side of the fraction line. So, we rewrite \(x^{-9s}\) using the Negative Exponent Rule as follows: \[x^{-9s} = \frac{1}{x^{9s}}\].

Key Concepts

Understanding the Power of a Power Rule

The power of a power rule in exponentiation is a fundamental concept that simplifies expressions with exponents. Whenever you have an expression like

• \((a^m)^n\),

you're dealing with a power raised to another power.
According to the rule, you multiply the exponents together. This means makes the calculation straightforward, transforming

• \(a^{m \times n}\).

In our exercise, we see

• \((x^{-3s})^3\).

Here, we apply the rule by multiplying

• \(-3s\) by the exponent
• 3, resulting in \(x^{-3s \times 3}\).

Now, this calculates to \(x^{-9s}\).
The power of a power rule helps in simplifying expressions, making complicated calculations much easier and manageable.
It's a staple in mathematics that you will use frequently as you advance in algebra.

Mastering the Negative Exponent Rule

The negative exponent rule is a concept that alters the position of the base relative to the fraction line. When an exponent is negative, it signifies that the base is on the incorrect side of the fraction line.
In general, any base raised to a negative exponent is equivalent to its reciprocal with a positive exponent. Mathematically, this means:

• \(a^{-n} = \frac{1}{a^n}\).

So, let's apply this to our specific case in the exercise:

• The expression \(x^{-9s}\) needs simplification.
• According to the rule, this transforms to \(\frac{1}{x^{9s}}\).

The negative exponent indicates a switch from a large power to a fraction.
Utilizing this rule helps in restructuring expressions to make calculations easier.
Understanding and applying the negative exponent rule is crucial for effective algebraic manipulation.

Simplifying Expressions with Exponents

Simplifying expressions is all about reducing them to their most efficient and manageable form. When working with exponents, simplifying involves using rules like the power of a power rule and the negative exponent rule.
It’s essential to know which rules apply:

• When dealing with powers, you should look for possibilities to apply the power of a power rule.
• Similarly, negative exponents can be simplified by converting them to positive exponents.

For our exercise: We began with

• \((x^{-3s})^3\)
• then simplified to \(x^{-9s}\),
• finally applying the negative exponent rule to get \(\frac{1}{x^{9s}}\).

The entire simplification process makes complex expressions look and behave simpler.
Remember, simplifying not only makes interpretation easy but also assists in solving equations efficiently.
Mastering these rules will be invaluable as you progress in mathematics, helping you tackle even more intricate problems with confidence.