Problem 49 - Algebra for College Students

Question

Use scientific notation and the properties of exponents to help you perform the following operations. \((90,000)^{\frac{3}{2}}\)

Step-by-Step Solution

Answer

\((90,000)^{\frac{3}{2}} = 2.7 \times 10^7\).

Step 1: Rewrite in Scientific Notation

First, rewrite 90,000 in scientific notation. The number 90,000 can be expressed as \(9 \times 10^4\) because it has four zeros and starts with 9.

Step 2: Apply Exponent to Scientific Notation

Now apply the exponent \(\frac{3}{2}\) to the number in scientific notation. This means you will calculate \((9 \times 10^4)^{\frac{3}{2}}\).

Step 3: Distribute the Exponent

Use the property \((ab)^n = a^n \cdot b^n\) to distribute the exponent to the base 9 and \(10^4\). This gives you \(9^{\frac{3}{2}} \cdot (10^4)^{\frac{3}{2}}\).

Step 4: Calculate \(9^{\frac{3}{2}}\)

Find \(9^{\frac{3}{2}}\) by rewriting it as \((9^1)^\frac{3}{2} = (9^\frac{1}{2})^3\). You know that \(9^\frac{1}{2} = 3\) because 9 is a perfect square. Therefore, \(3^3 = 27\).

Step 5: Calculate \((10^4)^{\frac{3}{2}}\)

Use the rule \((a^m)^n = a^{m \cdot n}\). Thus, \((10^4)^{\frac{3}{2}} = 10^{6}\).

Step 6: Multiply the Results

Combine the results from Steps 4 and 5. \(27 \cdot 10^6 = 2.7 \times 10^7\). This is the result of the original expression.

Key Concepts

Properties of Exponents

Exponents come with certain rules known as properties, which make simplifying expressions much easier. These rules are very helpful when you're dealing with more complex expressions involving powers. In the problem above, we dealt with the expression \[(90,000)^{\frac{3}{2}}\]by rewriting it using the properties of exponents.
One key property is that when an entire expression, like \((ab)^n\), is raised to a power, you can apply the power separately to each part of the expression: \[(ab)^n = a^n \cdot b^n\]Understanding this rule allows us to distribute the exponent \(\frac{3}{2}\) to both terms in \[(9 \times 10^4)^{\frac{3}{2}},\]turning it into \[9^{\frac{3}{2}} \cdot (10^4)^{\frac{3}{2}}.\]
Another crucial property is how to handle powers of powers. If you have an expression like \((a^m)^n\), you can simplify it using: \[(a^m)^n = a^{m \cdot n}\]In the problem, this is used to transform \((10^4)^{\frac{3}{2}}\) into \(10^{4 \times \frac{3}{2}} = 10^6.\) Keeping these properties in mind simplifies complex expressions enormously.

Numerical Operations

Numerical operations involve the steps of simplifying numerical expressions, often through addition, subtraction, multiplication, or division. In this context, when expressions involve exponents, numerical operations become more about applying the properties of exponents.
For \[9^{\frac{3}{2}},\]we simplify this operation by recognizing patterns or familiar squares. It's helpful to break down the process into smaller steps—you first find the square root of 9, which is 3, and then cube that result, yielding 27.
Similarly, for our expression \[(10^4)^{\frac{3}{2}},\]we can multiply the exponents, keeping in mind the power of a power property and recognizing that \[10^{6}\]is a straightforward expression when applying numerical operations. Together, these simplified results allowed us to multiply them directly to achieve the final result \(2.7 \times 10^7\).
Performing these numerical operations smoothly and correctly is essential for simplifying expressions correctly and efficiently.

Mathematical Expression Evaluation

Evaluating mathematical expressions is the process of finding their value using various mathematical rules and computations. Before you evaluate any expression, you need a clear path for simplification, often involving rewriting numbers and operations.
In this exercise, we started by writing 90,000 in scientific notation as \[9 \times 10^4,\]preparing it for further evaluation with the exponent \(\frac{3}{2}.\) Preceding with strategic application of exponent properties and numeric operations provided a streamlined path to simplify the given expression step by step.
After distributing and calculating the exponents separately—namely \[9^{\frac{3}{2}} = 27 \]and \[(10^4)^{\frac{3}{2}} = 10^6,\]we combined the results. The final step multiplied these values to obtain \(27 \cdot 10^6,\) eventually expressed in scientific notation as \(2.7 \times 10^7.\)
This process of evaluation not only highlights the potency of proper method application but also reinforces the connection between understanding abstract mathematical principles and arriving at a correct answer.