Problem 21 - Algebra for College Students

Question

Simplify each numerical expression. \(\left(3^{-1}\right)^{-3}\)

Step-by-Step Solution

Answer

The expression simplifies to 27.

Step 1: Apply the Power of a Power Property

The expression \((3^{-1})^{-3}\) can be simplified using the power of a power property. This states that \((a^m)^n = a^{m \times n}\). Therefore, simplify by multiplying the exponents: \(-1\) and \(-3\). You get \(3^{-1 \times (-3)} = 3^3\).

Step 2: Calculate the Final Power

Now that we have simplified the exponent, we compute \(3^3\). This means \(3\times 3\times 3\). Calculate it: \(3 \times 3 = 9\) and \(9 \times 3 = 27\). Therefore, \(3^3 = 27\).

Key Concepts

Power of a Power Property

The power of a power property is a fundamental rule in exponentiation that involves taking an exponent raised to yet another exponent. The rule can be stated as follows: if you have \[ (a^m)^n = a^{m \times n} \]. When you see a problem like \( (3^{-1})^{-3} \), you can simplify it by multiplying the inner exponent \(-1\) by the outer exponent \(-3\).To apply this rule, - Identify the base and the exponents. Here, the base is \(3\), with an inner exponent of \(-1\) and an outer exponent of \(-3\). - Multiply the exponents: \(-1 \times (-3) = 3\).By multiplying these exponents, you effectively reduce the complexity of the expression, resulting in \(3^3\). This property is especially helpful when dealing with nested exponents, allowing you to consolidate multiple powers into a single exponential expression.

Negative Exponents

Negative exponents can often seem tricky but they are essentially an indication that the base is on the wrong side of a fraction. If you encounter \(a^{-n}\), it's equivalent to \(\frac{1}{a^n}\). This means you're dealing with a reciprocal. Consider \(3^{-1}\): - Here, it's the same as writing \(\frac{1}{3^1}\). So, \(3^{-1} = \frac{1}{3}\).- Negative exponents often appear in situations where we're inverting a base or where a fraction is needed.When they occur within a larger exponential operation, such as \((3^{-1})^{-3}\), use properties of exponents to clarify the expression before addressing the negative exponent. In such cases, always simplify the exponents first, before interpreting the negative sign.

Simplifying Expressions

Simplifying expressions involves reducing a complex structure into a more manageable one, often by applying mathematical properties. The objective is to make the expression easier to understand or compute. Here's how we can simplify exponential expressions using what we know:- **Apply Exponential Rules:** First, use rules like the power of a power property to consolidate exponents. For example, in \((3^{-1})^{-3}\), this is simplified using\(3^{-1 \times (-3)}\)to get \(3^3\).- **Calculate Powers:** Once simplification is applied, compute any remaining powers. Here we moved from \(3^3\) to calculating \(3 \times 3 \times 3\), resulting in \(27\).- **Check Work:** Make sure to double-check each step along the way to avoid any errors or misinterpretations, especially when exponents involve negatives. Simplifying expressions is not just about performing calculations but understanding the algebraic relationships between numbers and applying known properties to achieve a simpler form.