Problem 5 - Basic Engineering Mathematics

Question

Evaluate: \(\frac{\left(10^{2}\right)^{3}}{10^{4} \times 10^{2}}\)

Step-by-Step Solution

Answer

The evaluated expression is 1.

Step 1: Simplify the Power of a Power

We start by simplifying the numerator, which is \( (10^2)^3 \). To simplify a power of a power, multiply the exponents: \( 2 \times 3 = 6 \). Thus, \( (10^2)^3 = 10^6 \).

Step 2: Multiply the Denominators

Look at the denominator \( 10^4 \times 10^2 \). Using the property of exponents for multiplying, add the exponents: \( 4 + 2 = 6 \). Thus, \( 10^4 \times 10^2 = 10^6 \).

Step 3: Divide Using the Property of Exponents

Now divide the simplified numerator by the simplified denominator: \( \frac{10^6}{10^6} \). When dividing with the same base, subtract the exponents: \( 6 - 6 = 0 \). Therefore, \( \frac{10^6}{10^6} = 10^0 \).

Step 4: Evaluate the Result

Any number raised to the power 0 is 1. Therefore, \( 10^0 = 1 \).

Key Concepts

Power of a Power

The concept of "power of a power" in exponentiation can be tricky, but it is actually quite simple. When you have an exponent raised to another exponent, like \((10^2)^3\), you don't just multiply the numbers but multiply the exponents themselves. This property means that multiplying the exponent of the base \(10^2\) by the power \(3\) results in \(10^{(2 \times 3)} = 10^6\).

• This rule helps to simplify expressions with multiple layers of exponents.
• Remember to always multiply the exponents when you see a power of a power.

The result is an expression with a single exponent, making it much easier to manage in further calculations.

Multiplying Exponents

When multiplying numbers with the same base, it's important to add their exponents together. For example, if you have \(10^4 \times 10^2\), the base number \(10\) stays the same, and you simply add the exponents: \(4 + 2\). This gives you \(10^6\), simplifying your expression.

• This is a straightforward and powerful rule, especially useful in algebra.
• Make sure both bases are the same before you proceed to add the exponents.

By following this rule, you can easily break down complex multiplication problems involving powers.

Dividing Exponents

Dividing exponents involves another key rule in exponentiation: subtracting the exponents. If you have something like \(\frac{10^6}{10^6}\), both numbers have the same base \(10\), so you subtract the exponents from each other: \(6 - 6\). This calculation leaves you with \(10^0\).

• This rule applies whenever you divide terms with the same base.
• Always subtract the exponent of the denominator from the exponent of the numerator.

Understanding this property simplifies divisions in expressions involving exponents.

Zero Exponent Property

The zero exponent property is a simple yet often surprising rule. Any number raised to the power of zero equals one. For example, \(10^0 = 1\). This might seem strange at first but makes mathematical sense. If we apply all the previous rules for exponent operations, we end up with 10 repeated zero times, which conceptually results in one.

• This rule is universal: any non-zero base raised to zero becomes one.
• Be careful: zero raised to any positive power is zero, and zero raised to zero is undefined.

This property helps to simplify and solve problems, reducing even seemingly complex expressions to a simple one.