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Answer :
Option D
The fraction equal to the fraction [tex]\left(\frac{4}{5}\right)^{6}[/tex] is [tex]\frac{4^{6}}{5^{6}}[/tex]
Solution:
Need to check which of the given option is equal to fraction [tex]\left(\frac{4}{5}\right)^{6}[/tex]
Lets look at one of the law of exponents which is useful in our case
[tex]\text { For any fraction }\left(\frac{a}{b}\right)^{m}=\frac{a^{m}}{b^{m}} \text { where } a \text { and } b \text { are integers }[/tex]
In our case a = 4 and b = 5 and m = 6
On applying the above law of exponents we get
[tex]\left(\frac{4}{5}\right)^{6}=\frac{4^{6}}{5^{6}}[/tex]
Also we can do it in another way:
[tex]\left(\frac{4}{5}\right)^{6}=\frac{4}{5} \times \frac{4}{5} \times \frac{4}{5} \times \frac{4}{5} \times \frac{4}{5} \times \frac{4}{5}=\frac{4 \times 4 \times 4 \times 4 \times 4 \times 4}{5 \times 5 \times 5 \times 5 \times 5 \times 5}=\frac{4^{6}}{5^{6}}[/tex]
Hence we can say that [tex]\left(\frac{4}{5}\right)^{6}=\frac{4^{6}}{5^{6}}[/tex] and correct option is D
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Final answer:
The correct answer is option D: 4⁶/5⁶, which represents the fraction (4/5) raised to the sixth power, with both the numerator and denominator being raised individually to the power of 6.
Explanation:
To find which of the following is equal to the fraction (4/5)⁶, it's important to understand how exponents work with fractions.
The exponent outside the parenthesis indicates that the fraction inside should be multiplied by itself six times. The correct procedure is to raise both the numerator and the denominator to the power of 6. Therefore, the correct simplification would be option D: 4⁶/5⁶. This means 4 raised to the 6th power divided by 5 raised to the 6th power.
The other options can be quickly dismissed:
Option A, 6(4/5), represents six times the original fraction, not the fraction raised to a power.
Option B, 24/30, is not a power but a simplified fraction equivalent to 4/5.
Option C, 4⁶/5, simply raises the numerator to the power of 6, but ignores the exponent for the denominator.
