Problem 50 - Algebra for College Students

Question

Use scientific notation and the properties of exponents to help you perform the following operations. \((8000)^{\frac{2}{3}}\)

Step-by-Step Solution

Answer

The result of \((8000)^{\frac{2}{3}}\) is 400.

Step 1: Write in Scientific Notation

Start by expressing 8000 in scientific notation. Since 8000 can be written as \(8 \times 10^3\), use this representation for further operations.

Step 2: Apply the Power of a Product Rule

The power of a product rule states \((ab)^n = a^n \times b^n\). Apply this rule to \((8 \times 10^3)^{\frac{2}{3}}\), which becomes \(8^{\frac{2}{3}} \times (10^3)^{\frac{2}{3}}\).

Step 3: Calculate \(8^{\frac{2}{3}}\)

The expression \(8^{\frac{2}{3}}\) can be evaluated by determining the cube root of 8, which is 2, and then squaring it, resulting in 4.

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

Apply the power of a power rule \((a^m)^n = a^{m \times n}\) to \((10^3)^{\frac{2}{3}}\), resulting in \(10^{3 \times \frac{2}{3}} = 10^2\).

Step 5: Combine the Results

Multiply the results from steps 3 and 4. Therefore, \(8^{\frac{2}{3}} \times (10^3)^{\frac{2}{3}} = 4 \times 10^2 = 400\).

Key Concepts

Properties of Exponents

Understanding the properties of exponents is crucial when working with powers in mathematics. These properties allow us to simplify complex expressions and perform calculations more efficiently. Here are some key properties:

• Product of Powers: When multiplying two exponents with the same base, you add the exponents, i.e., \(a^m \times a^n = a^{m+n}\).
• Quotient of Powers: Dividing two exponents with the same base means subtracting the exponents, i.e., \(a^m / a^n = a^{m-n}\).
• Power of a Power: To raise an exponent to another power, multiply the exponents, i.e., \((a^m)^n = a^{m \times n}\).
• Zero Exponent: Any base raised to the zero power is 1, i.e., \(a^0 = 1\) if \(a eq 0\).
• Negative Exponent: A negative exponent indicates the reciprocal, i.e., \(a^{-n} = 1/a^n\).

By mastering these properties, solving exercises like the one given becomes straightforward and less daunting. You can easily simplify and manipulate expressions using these rules.

Power of a Product Rule

The power of a product rule is an essential exponentiation law that simplifies expressions involving products raised to a power. According to this rule,

• \((ab)^n = a^n \times b^n\)

This means that each factor in the product is separately raised to the given power. For example, applying this rule to \((8 \times 10^3)^{\frac{2}{3}}\) results in \(8^{\frac{2}{3}} \times (10^3)^{\frac{2}{3}}\).
This concept allows us to break down more extensive expressions into manageable portions. Applying the power of a product rule simplifies calculations, particularly when working with larger numbers or expressions in scientific notation.

Scientific Notation in Mathematics

Scientific notation is a method used to express very large or very small numbers more conveniently. It allows numbers to be written as the product of a number between 1 and 10 and a power of ten.

• For instance, 8000 can be written in scientific notation as \(8 \times 10^3\). Here, 8 is a number between 1 and 10, and \(10^3\) indicates how many times 10 should be multiplied by itself.

Scientific notation simplifies computation and comparison in many fields, especially when handling data on a grand scale. When we want to perform operations like multiplication or exponentiation on these numbers, scientific notation makes the process clearer and more manageable. Expressing numbers this way is particularly helpful when using properties of exponents to simplify expressions.

Exponentiation

Exponentiation is a mathematical operation involving numbers known as bases and powers (or exponents). The expression \(a^n\) means that the base \(a\) is multiplied by itself \(n\) times.

• For example, \(2^3 = 2 \times 2 \times 2 = 8\).

Exponentiation can extend beyond whole numbers to include fractions, negative numbers, and even variables. Understanding how exponentiation interacts with other mathematical principles, such as scientific notation and exponent rules, is vital for solving problems involving power.
In our initial exercise, exponentiation is used in the expression \((8000)^{\frac{2}{3}}\). This fraction exponent suggests a combination of roots and powers, further exemplifying the operation's versatility.

Power of a Power Rule

The power of a power rule is an exponent rule that involves raising an already-expressed power to another power. The rule states:

• \((a^m)^n = a^{m \times n}\)

With this rule, you multiply the exponents to get a single power. For example, in the expression \((10^3)^{\frac{2}{3}}\), applying the power of a power rule results in \(10^{3 \times \frac{2}{3}} = 10^2\). This shows the simplification of taking a power of a power without having to manually calculate intermediate steps.
This rule is particularly useful when dealing with nested exponentiation, helping to streamline complex calculations and making them easier to evaluate.