Error Analysis: Correctly Equating Exponents Using the Power Property
Grade 9 students in California Reveal Math Algebra 1 learn to identify and correct the most common error made when using the Power Property of Equality: accidentally moving the base into the exponent. When both sides share the same base b, you simply set the exponents equal — you do not move the base. For example, 2^(7x)=2^2 should become 7x=2 (giving x=2/7), NOT 7^x=2. Students also see errors like replacing 3^(5x)=3^4 with 5^x=4 (should be 5x=4, x=4/5) and learn to always substitute their solution back to verify.
Key Concepts
When both sides of an equation share the same base $b$ (where $b 0$ and $b \neq 1$), the Power Property of Equality says to set the entire exponents equal to each other — not to move the base into the exponent expression.
$$b^{\text{exponent} 1} = b^{\text{exponent} 2} \implies \text{exponent} 1 = \text{exponent} 2$$.
Common Questions
What is the Power Property of Equality?
The Power Property of Equality states that when two exponential expressions have the same base b, you set their exponents equal: if b^(expr1) = b^(expr2), then expr1=expr2. The base disappears and only the exponents remain.
What is the most common error when using the Power Property of Equality?
The most common error is moving the base into the exponent expression. For example, writing 2^(7x)=2^2 as 7^x=2 is wrong. The correct step is simply 7x=2, giving x=2/7.
How do you correctly solve 3^(5x)=3^4?
Since both sides have base 3, set the exponents equal: 5x=4. Divide by 5: x=4/5. The common error would be writing 5^x=4, which changes the problem entirely.
How do you check your answer after applying the Power Property?
Substitute your solution back into the original equation and confirm both sides are equal. This catches errors made during the exponent-equating step.
Can you identify the error in x^(-3)=7 as a solution step for 5^(x-3)=5^7?
The error is treating the base 5 as if it becomes the variable. The correct step is setting exponents equal: x-3=7, giving x=10.
Which unit covers this error analysis skill?
This skill is from Unit 7: Exponents and Roots in California Reveal Math Algebra 1, Grade 9.