4. Which expressions can be rewritten as 1 over the base raised to a positive exponent? (2 points)
start fraction b superscript negative 2 baseline over b superscript 4 baseline end fraction
start fraction w superscript 5 baseline over w superscript negative 3 baseline end fraction
Start Fraction 5 superscript 3 baseline over 5 superscript 5 baseline end fraction
Start Fraction 4 superscript 7 baseline over 4 superscript 7 baseline end fraction
Only one expression can be rewritten as 1 over the base raised to a positive exponent:
-start fraction b superscript negative 2 baseline over b superscript 4 baseline end fraction can be rewritten as 1 over b squared.
To rewrite an expression as 1 over the base raised to a positive exponent, we need to follow the rule that a negative exponent can be rewritten as the reciprocal of the base raised to the positive exponent.
Let's go through each expression:
1. start fraction b superscript negative 2 baseline over b superscript 4 baseline end fraction:
This expression can be rewritten as 1 over b raised to the absolute value of the negative exponent. Therefore, it becomes 1 over b squared.
2. start fraction w superscript 5 baseline over w superscript negative 3 baseline end fraction:
This expression can be rewritten as 1 over w raised to the absolute value of the negative exponent. Therefore, it becomes 1 over w cubed.
3. Start fraction 5 superscript 3 baseline over 5 superscript 5 baseline end fraction:
This expression does not have a negative exponent, so it cannot be rewritten as 1 over the base raised to a positive exponent.
4. Start fraction 4 superscript 7 baseline over 4 superscript 7 baseline end fraction:
This expression does not have a negative exponent, so it cannot be rewritten as 1 over the base raised to a positive exponent.
Therefore, the expressions that can be rewritten as 1 over the base raised to a positive exponent are:
- 1 over b squared
- 1 over w cubed
To determine which expressions can be rewritten as 1 over the base raised to a positive exponent, we need to identify the expressions where the base appears as a negative exponent or where there are negative exponents in the denominator. Let's examine each expression:
1. To rewrite the expression start fraction b superscript negative 2 baseline over b superscript 4 baseline end fraction as 1 over the base raised to a positive exponent, we can simplify it by applying the rule that a negative exponent in the numerator becomes a positive exponent in the denominator. We have:
start fraction b superscript negative 2 baseline over b superscript 4 baseline end fraction = 1 over b superscript 2 baseline times b superscript negative 4 baseline
Now, using another exponent rule, we combine the two terms in the denominator by subtracting the exponents:
= 1 over b superscript (2 - 4) baseline
Simplifying further:
= 1 over b superscript negative 2 baseline
Therefore, the first expression can be rewritten as 1 over the base raised to a positive exponent.
2. The expression start fraction w superscript 5 baseline over w superscript negative 3 baseline end fraction can be simplified following similar steps. We rewrite it as:
start fraction w superscript 5 baseline over w superscript negative 3 baseline end fraction = w superscript 5 baseline times w superscript negative 3 baseline
Using the rule for multiplying exponents:
= w superscript (5 + (-3)) baseline
Simplifying further:
= w superscript 2 baseline
Therefore, the second expression cannot be rewritten as 1 over the base raised to a positive exponent.
3. The expression Start Fraction 5 superscript 3 baseline over 5 superscript 5 baseline end fraction can be simplified by subtracting the exponents:
Start Fraction 5 superscript 3 baseline over 5 superscript 5 baseline end fraction = 5 superscript (3 - 5) baseline
Simplifying further:
= 5 superscript (-2) baseline
Therefore, the third expression can be rewritten as 1 over the base raised to a positive exponent.
4. The expression Start Fraction 4 superscript 7 baseline over 4 superscript 7 baseline end fraction has the same exponent in both the numerator and denominator, so we can simplify it to:
Start Fraction 4 superscript 7 baseline over 4 superscript 7 baseline end fraction = 1
Therefore, the fourth expression can be rewritten as 1 over the base raised to a positive exponent.
To summarize, the expressions that can be rewritten as 1 over the base raised to a positive exponent are the first expression (b^(-2)/b^4), the third expression (5^3/5^5), and the fourth expression (4^7/4^7).