Simplify a67/b34 .(1 point)
Responses
a67/b34
Start Fraction a superscript 67 baseline over b superscript 34 baseline End Fraction
a33
a superscript 33 baseline
23
23
ab33
When we simplify the expression \( \frac{a^{67}}{b^{34}} \), it is important to recognize that there are no direct simplifications that can be made unless we have additional information about the relationship between \( a \) and \( b \). However, I can explain what each of the provided options means:
1. \( \frac{a^{67}}{b^{34}} \) – This is the original expression as provided, with \( a \) raised to the 67th power divided by \( b \) raised to the 34th power.
2. Start Fraction \( a^{67} \) over \( b^{34} \) End Fraction – This is a verbal description of the original expression using the words "Start Fraction" and "End Fraction" to indicate the format of a fraction, which is not a simplification.
3. \( a^{33} \) – This would be the result if you could cancel out 34 powers of \( a \) with the 34 powers of \( b \) from the denominator assuming \( a \) equals \( b \). However, since we have no information as to whether \( a \) and \( b \) are the same, this is not applicable.
4. 23 – This has no relation to the original expression and cannot be a simplification of \( \frac{a^{67}}{b^{34}} \) without additional context.
5. \( ab^{33} \) – This seems like an attempt at simplification, but it is incorrect. It would imply that \( a^{67} \) can somehow be represented as \( a \times b^{33} \), which is not true.
Without any further information on the relationship between \( a \) and \( b \), the expression \( \frac{a^{67}}{b^{34}} \) cannot be simplified further and remains as it is.
Apply the Quotient Rule of Integer Exponents to simplify 2365÷2332 .(1 point)
Responses
97
97
2397
23 superscript 97 baseline
133
1 superscript 33 baseline
2333
The Quotient Rule of Integer Exponents states that for any nonzero base \( a \) and integers \( m \) and \( n \), the quotient \( a^m ÷ a^n = a^{m-n} \).
Applying this rule to the expression \( 23^{65} ÷ 23^{32} \), we subtract the exponents:
\( 23^{65} ÷ 23^{32} = 23^{65-32} = 23^{33} \).
So the correct simplified form of the expression is \( 23^{33} \).
Simplify this expression using the Quotient Rule of Integer Exponents: 1253÷753 .(1 point)
Responses
12106
12 superscript 106 baseline
5106
5 superscript 106 baseline
553
5 superscript 53 baseline
1253÷753
The given expression is \( 12^{53} ÷ 7^{53} \). Unfortunately, since the bases are different (12 and 7), the Quotient Rule of Integer Exponents, which states that \( a^m ÷ a^n = a^{m-n} \) for the same nonzero base \( a \) and any integers \( m \) and \( n \), cannot be directly applied here.
However, \( 12^{53} ÷ 7^{53} \) can be rewritten by factoring out the exponents since they are equal:
\( 12^{53} ÷ 7^{53} = \left(\frac{12}{7}\right)^{53} \)
Since there is no simplification that can reduce \( \frac{12}{7} \), the expression remains as:
\( \left(\frac{12}{7}\right)^{53} \)
This cannot be simplified any further without additional context or instructions. Consequently, neither of the provided responses (12^106, 5^106, 5^53, \( 12^{53} ÷ 7^{53} \)) are correct simplifications of the initial expression. If you are expected to find a simplified form, it would be appropriate to state it as \( \left(\frac{12}{7}\right)^{53} \). If the instructions are asking for an exponent form with a single base, the original bases do not allow for this since they are different and cannot be combined using the Quotient Rule of Integer Exponents.
Which are the correct steps to follow to simplify an exponential expression using the Quotient Rule of Integer Exponents?(1 point)
Responses
Check if the bases are different; if they are, subtract the exponents of the numerator and denominator; and write the resulting expression in exponential form.
Check if the bases are different; if they are, subtract the exponents of the numerator and denominator; and write the resulting expression in exponential form.
Check if the bases are the same; if they are, subtract the exponents of the numerator and denominator (leave as-is otherwise); and write the resulting expression in exponential form.
Check if the bases are the same; if they are, subtract the exponents of the numerator and denominator (leave as-is otherwise); and write the resulting expression in exponential form.
Check if the bases are the same; if they are, add the exponents of the numerator and denominator (leave as-is otherwise); and write the resulting expression in exponential form.
Check if the bases are the same; if they are, add the exponents of the numerator and denominator (leave as-is otherwise); and write the resulting expression in exponential form.
Subtract the exponents of the numerator and denominator; and write the resulting expression in exponential form.
The correct steps to simplify an exponential expression using the Quotient Rule of Integer Exponents are:
- Check if the bases are the same; if they are, subtract the exponents of the numerator and denominator (leave as-is otherwise); and write the resulting expression in exponential form.
- Check if the bases are the same; if they are, subtract the exponents of the numerator and denominator (leave as-is otherwise); and write the resulting expression in exponential form.
The Quotient Rule can only be applied when the bases of the exponential expressions are the same. When that is the case, you subtract the exponent in the denominator from the exponent in the numerator. Adding the exponents is not the Quotient Rule but the Product Rule for exponents.
The correct process then would be:
1. Verify that the bases are the same.
2. If the bases are the same, subtract the exponent of the denominator from the exponent of the numerator to get the new exponent.
3. Write the base raised to the resulting exponent.
For example, if you have \( a^m ÷ a^n \) and \( a \) is the same in both the numerator and the denominator, then the simplified form would be \( a^{m-n} \). If the bases are not the same, the Quotient Rule does not apply, and the expression should be left as is or simplified by factoring if possible.