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Chapter 5: Problem 16

Simplify each expression. \(\left(\frac{3}{8}\right)^{0}\)

Short Answer

Expert verified
\(\left(\frac{3}{8}\right)^{0} = 1\)

Step by step solution

01

Identify the Power of Zero Rule

The expression given is \(\left(\frac{3}{8}\right)^{0}\). Recall the rule that any non-zero base raised to the power of zero equals 1. This rule applies because multiplying a number by itself zero times (essentially, not multiplying it at all) results in 1.
02

Apply the Power of Zero Rule

Using the rule from Step 1, simplify \(\left(\frac{3}{8}\right)^{0}\) to 1. Since the base \(\frac{3}{8}\) is not zero, the power of zero results in the value of 1.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Power of Zero Rule
The Power of Zero Rule in mathematics states that any non-zero number raised to the power of zero is equal to 1. This might seem a bit strange at first, but it makes sense when you think about it in terms of exponents. An exponent signifies how many times to multiply the base by itself. So, when the exponent is zero, you're essentially not multiplying the number at all. That's why it defaults to 1.
  • Non-zero base to power zero: Always equals 1.
  • Zero base: The case where the base itself is zero can be tricky. Remember, the rule applies to non-zero numbers.

In our exercise, \left(\frac{3}{8}\right)^{0}\ is such a case. The base is \(\frac{3}{8}\), which is a non-zero fraction. Applying the Power of Zero Rule, the expression simplifies directly to 1.
Exponentiation
Exponentiation is a mathematical operation involving two numbers: the base and the exponent. The base is the number being multiplied, and the exponent tells us how many times the base multiplies itself. For instance, in the expression \(a^n\), \(a\) is the base and \(n\) is the exponent.
  • If \(n = 2\), then \(a^n = a \times a\).
  • If \(n = 3\), then \(a^n = a \times a \times a\).

For an exponent of zero, though, things change slightly. As we learned with the Power of Zero Rule, \(a^0 = 1\) for any \(a eq 0\). This is an essential tool for simplifying expressions in algebra and beyond.
Fractional Base
A fractional base means that the base of your expression is a fraction. This doesn't change the rules of exponents, but it might look a bit different. In our case, we have \(\left(\frac{3}{8}\right)^{0}\).
  • Base as a fraction: When the base is a fraction, treat it like any other number in terms of the power rules.
  • Fractional base to zero power: Just like with whole numbers, any non-zero fraction raised to the power of zero is always 1.

Even though fractions can sometimes feel more complex, they follow the same exponent rules. So in the problem provided, \(\left(\frac{3}{8}\right)^{0} = 1\), which highlights the elegance of the Power of Zero Rule across different types of numbers.

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