The image shows a practice packet for 8th-grade math, covering several topics. Below, I will address the visible problems and explain the solutions step by step.
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1. Rational vs. Irrational Numbers
#### Problem:
Identify the following numbers as RATIONAL or IRRATIONAL.
- Number 1: \( \frac{8}{9} \)
- Number 2: \( \pi \)
#### Solution:
-
Number 1: \( \frac{8}{9} \)
- A rational number is any number that can be expressed as the ratio of two integers (i.e., in the form \( \frac{p}{q} \), where \( p \) and \( q \) are integers and \( q \neq 0 \)).
- \( \frac{8}{9} \) is already in the form of a fraction of two integers.
- Therefore, \( \frac{8}{9} \) is
RATIONAL.
-
Number 2: \( \pi \)
- \( \pi \) (pi) is a well-known mathematical constant approximately equal to 3.14159.
- \( \pi \) cannot be expressed as a ratio of two integers. It is a non-repeating, non-terminating decimal.
- Therefore, \( \pi \) is
IRRATIONAL.
#### Final Answer:
\[
\boxed{\text{Rational, Irrational}}
\]
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2. Exponent Properties
#### Problem:
Simplify the given expressions using exponent properties.
- Expression 11: \( \left( \frac{10}{c} \right)^2 \)
- Expression 12: \( 2x^0 \)
- Expression 13: \( c^3 \cdot c^7 \)
#### Solution:
-
Expression 11: \( \left( \frac{10}{c} \right)^2 \)
- Use the property \( \left( \frac{a}{b} \right)^n = \frac{a^n}{b^n} \).
- Here, \( a = 10 \), \( b = c \), and \( n = 2 \).
- Simplify: \( \left( \frac{10}{c} \right)^2 = \frac{10^2}{c^2} = \frac{100}{c^2} \).
-
Expression 12: \( 2x^0 \)
- Use the property \( x^0 = 1 \) for any nonzero \( x \).
- Simplify: \( 2x^0 = 2 \cdot 1 = 2 \).
-
Expression 13: \( c^3 \cdot c^7 \)
- Use the property \( a^m \cdot a^n = a^{m+n} \).
- Here, \( a = c \), \( m = 3 \), and \( n = 7 \).
- Simplify: \( c^3 \cdot c^7 = c^{3+7} = c^{10} \).
#### Final Answers:
\[
\boxed{\frac{100}{c^2}, 2, c^{10}}
\]
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3. Functions: Rates of Change & Initial Values
#### Problem:
Determine the rate of change and initial value for the given function.
- Function: Not explicitly provided in the image.
#### Explanation:
To solve this part, we would need the specific function. Since the function is not provided in the image, I cannot determine the rate of change and initial value. If you provide the function, I can help solve it.
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Summary of Solutions
1.
Rational vs. Irrational Numbers:
- \( \frac{8}{9} \): Rational
- \( \pi \): Irrational
2.
Exponent Properties:
- \( \left( \frac{10}{c} \right)^2 = \frac{100}{c^2} \)
- \( 2x^0 = 2 \)
- \( c^3 \cdot c^7 = c^{10} \)
3.
Functions: Rates of Change & Initial Values:
- Requires the specific function to solve.
If you have additional details or need further clarification, feel free to ask!
Parent Tip: Review the logic above to help your child master the concept of 8th grade math common core worksheet.
