Question

Question 1:
Powers of 3 Value
3^3 27
3^2 9
3^1 3
3^0
3^-1
3^-2 —10
3^-3
What is the pattern as the exponents decrease? ____
What is the value of each variable?
a =
b =
c =

Question 2:
What is the value of the expression (-5)^-3 ?
1. Apply the negative exponent rule: (-5)^-3= 1/(-5)^3
2. Expand the power: (-5)^-3 =1/(5) = 1/(-5)(-5)(-5)
3. Simplify : (-5)^-3 = 1/(-5)^3 =1/ (-5)(-5)(-5)
What is the value of x?
(1 point)
• x = 15
0 x = 125
* 2=－125
* 2＝－15

Question 3:

Iris found a value of 1/64 when she evaluated an expression. Which could have been the expression Iris evaluated? Check the two that apply. (2 points)

2^-6

6^4

(1/4)^-3

(1/2)^-5

4^-3

Answer 1

Determine the value of the variable in the simplified expressions.
What is the value of x?
g^x + h -3 = 1/9^6 + 1/h^3

X=___
What is the value of y?

B^4c^-7 = b^y/c^7

Y=

Find the value of the expression
[ (3/5)^0) ] ^-2
(1 point)

Answer 2

### Question 1:

To find the missing values in the powers of 3 table, let's first identify the missing variable values (3^0, 3^-1, and 3^-3):

- \(3^0 = 1\)
- For \(3^{-1}\):
\[
3^{-1} = \frac{1}{3} \approx 0.333
\]
- For \(3^{-2}\):
\[
3^{-2} = \frac{1}{3^2} = \frac{1}{9} \approx 0.111
\]
- For \(3^{-3}\):
\[
3^{-3} = \frac{1}{3^3} = \frac{1}{27} \approx 0.037
\]

Now, filling in the values, we have:
- **3^0 = 1**
- **3^-1 = 1/3**
- **3^-2 = 1/9** (which is approximately 0.111)
- **3^-3 = 1/27** (which is approximately 0.037)

**Pattern as the exponents decrease:** As the exponents decrease, the values of \(3^x\) decrease, transitioning from larger positive numbers to fractions and finally approaching zero.

Values:
- a = 1 (which is \(3^0\))
- b = 1/3 (which is \(3^{-1}\))
- c = 1/27 (which is \(3^{-3}\))

### Question 2:

Now let's calculate the value of \((-5)^{-3}\):

1. Start with the negative exponent rule:
\[
(-5)^{-3} = \frac{1}{(-5)^3}
\]
2. Expand the power:
\[
(-5)^3 = -5 \cdot -5 \cdot -5 = -125
\]
So:
\[
(-5)^{-3} = \frac{1}{-125}
\]

Therefore, the value of **x** is:
- **x = -\frac{1}{125}** (none of the options are stated as fractions, but if considering positive or negative values, the closest would be \(x = -125\)).

### Question 3:

To find which expressions can evaluate to \( \frac{1}{64} \):

1. **\(2^{-6}\)**:
\[
2^{-6} = \frac{1}{2^6} = \frac{1}{64}
\]
This is true.

2. **\(6^4\)**:
\[
6^4 = 1296 \quad (\text{not } \frac{1}{64})
\]

3. **\(\left(\frac{1}{4}\right)^{-3}\)**:
\[
\left(\frac{1}{4}\right)^{-3} = 4^3 = 64 \quad (\text{not } \frac{1}{64})
\]

4. **\(\left(\frac{1}{2}\right)^{-5}\)**:
\[
\left(\frac{1}{2}\right)^{-5} = 2^5 = 32 \quad (\text{not } \frac{1}{64})
\]

5. **\(4^{-3}\)**:
\[
4^{-3} = \frac{1}{4^3} = \frac{1}{64}
\]
This is true.

Thus, the two expressions that Iris could have evaluated are:
- **\(2^{-6}\)**
- **\(4^{-3}\)**

Answer 3

Let's go through each part of your questions step by step.

### Part 1: Determine the value of \( x \)

Given the equation:
\[
g^x + h - 3 = \frac{1}{9^6} + \frac{1}{h^3}
\]

To solve for \( x \), we'll first simplify the right-hand side.

1. Simplify \( \frac{1}{9^6} \):
- Since \( 9 = 3^2 \), we have:
\[
9^6 = (3^2)^6 = 3^{12}
\]
So:
\[
\frac{1}{9^6} = \frac{1}{3^{12}} = 3^{-12}
\]

2. Rewrite the equation:
\[
g^x + h - 3 = 3^{-12} + \frac{1}{h^3}
\]

If we need to isolate \( g^x \), we can rearrange the equation:
\[
g^x = 3^{-12} + \frac{1}{h^3} + 3
\]

However, without specific values for \( g \), \( h \), or further information or constraints, we can't solve directly for \( x \). We can only express \( x \) in terms of the variables and constants:
- \( g^x = 3^{-12} + \frac{1}{h^3} + 3 \) to solve for \( x \) if \( g \) is known.

### Part 2: Determine the value of \( y \)

Given the equation:
\[
B^4c^{-7} = \frac{b^y}{c^7}
\]

From this equation, we notice that \(B\) and \(b\) might be the same base. Assuming they are indeed the same (let’s denote them as \( b \)), the equation can be rewritten as:
\[
b^4c^{-7} = \frac{b^y}{c^7}
\]

This implies:
1. The coefficients of \( b \):
\[
4 = y
\]
2. The coefficients of \( c \) are equal on both sides since \( c^{-7} = c^{-7} \) (the same for both sides).

Thus,
\[
y = 4
\]

### Part 3: Find the value of the expression

Evaluate the expression:
\[
\left( \left( \frac{3}{5} \right)^0 \right)^{-2}
\]

1. First, simplify \( \left( \frac{3}{5} \right)^0 \):
- Any non-zero number raised to the power of 0 equals 1, so:
\[
\left( \frac{3}{5} \right)^0 = 1
\]

2. Now evaluate the outer exponent:
\[
(1)^{-2} = 1
\]

Therefore, the value of the expression is:
\[
1
\]

### Summary of values:
- \( x \): Expressed in terms of variables, \( g^x = 3^{-12} + \frac{1}{h^3} + 3 \)
- \( y = 4 \)
- The value of the expression is \( 1 \)