Question

Vocabulary Terms - Define/Illustrate

What does an exponent tell you about the base?
What if the exponent = 0?
What if the exponent = negative?


5 Problems - SHOW ALL WORK! Expand out if needed.

7^7/(7^2x7^5)=






r-2st3u0=
8^0m^0n^-4* 9n=







t^-5/4^0t^4=
Why is anything to the zero power = 1 ?
Write out using division to show the reasoning.

Answer 1

### Vocabulary Terms

1. **Exponent**: An exponent tells you how many times the base is multiplied by itself. For example, in \( a^n \), \( a \) is the base and \( n \) is the exponent. It indicates repeated multiplication: \( a^n = a \times a \times a \ldots \) (n times).

2. **Exponent = 0**: When the exponent is 0, any non-zero base raised to the power of 0 equals 1. This means \( a^0 = 1 \) for any \( a \neq 0 \). This can be understood through the property of exponents, where dividing a base by itself (subtracting exponents) yields \( a^0 \).

3. **Exponent = Negative**: When the exponent is negative, it indicates the reciprocal of the base raised to the positive exponent. For instance, \( a^{-n} = \frac{1}{a^n} \). This means you take the reciprocal of the base raised to the absolute value of the exponent.

---

### Problems

**1.** \( \frac{7^7}{7^2 \times 7^5} \)

Using the property of exponents that says \( \frac{a^m}{a^n} = a^{m-n} \):

\[
= \frac{7^7}{7^{2+5}} \quad \text{(since } 7^2 \times 7^5 = 7^{2+5} = 7^7\text{)}
\]

\[
= \frac{7^7}{7^7} = 7^{7-7} = 7^0 = 1
\]

---

**2.** \( r^{-2}s t^{3}u^{0} \)

Using the property of exponents: \( a^0 = 1 \), we simplify \( u^0 \):

\[
= r^{-2} \times s \times t^{3} \times 1 = r^{-2} s t^{3}
\]

---

**3.** \( 8^{0} m^{0} n^{-4} \times 9n \)

Using \( a^{0} = 1 \):

\[
= 1 \times 1 \times n^{-4} \times 9n = n^{-4} \times 9n
\]

We can further simplify:

\[
= 9 \times \frac{n^{1}}{n^4} = 9n^{1-4} = 9n^{-3}
\]

---

**4.** \( \frac{t^{-5}}{4^0 t^4} \)

First, simplify \( 4^0 \):

\[
= \frac{t^{-5}}{1 \times t^4} = \frac{t^{-5}}{t^4}
\]

Using \( \frac{a^m}{a^n} = a^{m-n} \):

\[
= t^{-5-4} = t^{-9}
\]

---

**5. Why is anything to the zero power = 1?**

To show why \( a^0 = 1 \) using division, consider the exponent properties:

\[
a^n \div a^n = 1 \quad \text{(where } a \neq 0 \text{)}
\]

Now, using properties of exponents:

\[
\frac{a^n}{a^n} = a^{n-n} = a^0
\]

Thus, since \( \frac{a^n}{a^n} = 1 \):

\[
a^0 = 1
\]

This reasoning shows that when any non-zero number is raised to the power of zero, the result is 1.