Problem 242 In the following exercises, simp... [FREE SOLUTION]

91影视

Prealgebra

Lynn Marecek, MaryAnne Anthony-Smith

$Math Studyset 91影视 Explanations$ Math

2016 Edition

Chapter 10: Problem 242

In the following exercises, simplify. $$\text {(a)} -6^{0} \quad \text {(b)}-6^{1}$$

Short Answer

Expert verified

a) $ -1 $, b) $ -6 $

Step by step solution

Simplify $ -6^0 $

Any non-zero number raised to the power of zero is equal to 1. Therefore, $ -6^0 = -1 $.

Simplify $ -6^1 $

Any number raised to the power of one is equal to itself. Therefore, $ -6^1 = -6 $.

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Key Concepts

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

Understanding Exponents

Exponents are a way to express repeated multiplication of a number by itself. For example, if you have the expression $3^4$, it means multiplying 3 by itself 4 times: $3 \times 3 \times 3 \times 3$. An exponent has two parts:

The base: the number that is being multiplied.
The exponent: the small number written above and to the right of the base, indicating how many times the base is multiplied by itself.

Understanding exponents helps simplify long multiplication into more manageable chunks. For instance, instead of writing $3 \times 3 \times 3 \times 3$, we use $3^4$. This notation makes calculations and writing much simpler.

Power of Zero

One important rule to remember with exponents is the 'power of zero' rule. This rule states that any non-zero number raised to the power of zero equals 1. For instance, $5^0 = 1$.
This concept may seem strange at first, but it's a useful rule in mathematics. Here's how it works:

Consider $2^3 = 8$ and $2^2 = 4$. Dividing these gives $2^3 / 2^2 = 2^{3-2} = 2^1 = 2$.
Similarly, $2^2 / 2^2 = 2^{2-2} = 2^0 = 1$.

So, when you see an exponent of zero, remember that the value is always 1 regardless of what the base is (as long as the base is not zero). Hence, in our example, $-6^0 = -1$ is because the negative sign is not part of the base. The expression actually reads: $-(6^0)$.

Negative Numbers in Exponents

Negative numbers can sometimes be confusing when they're involved in exponent calculations. It's important to differentiate between negative bases and negative exponents. With our example, $-6^0$:

We know that any number raised to the zero power is 1.
Since the exponent applies to the 6 and not the negative sign, we first find $6^0 = 1$ and then apply the negative sign: $-6^0 = -1$.

For another part, like $-6^1$, it's a bit simpler:

Any number raised to the power of one is itself.
So, $-6^1$ just becomes $-6$.

When working with negative numbers and exponents, always pay attention to where the negative sign is placed. It can change the result significantly.
Understanding these basic rules will help simplify and correctly solve problems involving exponents in your math exercises.

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91影视

In the following exercises, simplify. $$\text {(a)} -6^{0} \quad \text {(b)}-6^{1}$$

Short Answer

Step by step solution

Simplify \( -6^0 \)

Simplify \( -6^1 \)

Key Concepts

Understanding Exponents

Power of Zero

Negative Numbers in Exponents

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