Join IDNLearn.com and start exploring the answers to your most pressing questions. Discover the information you need quickly and easily with our reliable and thorough Q&A platform.
Sagot :
Sure, let's prove the given equation step by step.
We start with the given expression:
[tex]\[ \frac{6^{x+2}-6^x}{6^{x+1}+6^x} \][/tex]
First, note that we can express [tex]\(6^{x+2}\)[/tex] and [tex]\(6^{x+1}\)[/tex] in terms of [tex]\(6^x\)[/tex]:
[tex]\[ 6^{x+2} = 6^x \cdot 6^2 \][/tex]
[tex]\[ 6^{x+1} = 6^x \cdot 6 \][/tex]
Substitute these into the given expression:
[tex]\[ \frac{6^x \cdot 6^2 - 6^x}{6^x \cdot 6 + 6^x} \][/tex]
Factor out [tex]\(6^x\)[/tex] from both the numerator and the denominator:
[tex]\[ \frac{6^x (6^2 - 1)}{6^x (6 + 1)} \][/tex]
Since [tex]\(6^x\)[/tex] is a common factor in both the numerator and the denominator, we can cancel it out:
[tex]\[ \frac{6^2 - 1}{6 + 1} \][/tex]
Next, evaluate the remaining terms:
[tex]\[ 6^2 = 36 \][/tex]
[tex]\[ 6 + 1 = 7 \][/tex]
So,
[tex]\[ \frac{36 - 1}{7} = \frac{35}{7} \][/tex]
Finally, divide 35 by 7:
[tex]\[ \frac{35}{7} = 5 \][/tex]
Thus, we have proven that:
[tex]\[ \frac{6^{x+2} - 6^x}{6^{x+1} + 6^x} = 5 \][/tex]
We start with the given expression:
[tex]\[ \frac{6^{x+2}-6^x}{6^{x+1}+6^x} \][/tex]
First, note that we can express [tex]\(6^{x+2}\)[/tex] and [tex]\(6^{x+1}\)[/tex] in terms of [tex]\(6^x\)[/tex]:
[tex]\[ 6^{x+2} = 6^x \cdot 6^2 \][/tex]
[tex]\[ 6^{x+1} = 6^x \cdot 6 \][/tex]
Substitute these into the given expression:
[tex]\[ \frac{6^x \cdot 6^2 - 6^x}{6^x \cdot 6 + 6^x} \][/tex]
Factor out [tex]\(6^x\)[/tex] from both the numerator and the denominator:
[tex]\[ \frac{6^x (6^2 - 1)}{6^x (6 + 1)} \][/tex]
Since [tex]\(6^x\)[/tex] is a common factor in both the numerator and the denominator, we can cancel it out:
[tex]\[ \frac{6^2 - 1}{6 + 1} \][/tex]
Next, evaluate the remaining terms:
[tex]\[ 6^2 = 36 \][/tex]
[tex]\[ 6 + 1 = 7 \][/tex]
So,
[tex]\[ \frac{36 - 1}{7} = \frac{35}{7} \][/tex]
Finally, divide 35 by 7:
[tex]\[ \frac{35}{7} = 5 \][/tex]
Thus, we have proven that:
[tex]\[ \frac{6^{x+2} - 6^x}{6^{x+1} + 6^x} = 5 \][/tex]
We appreciate your participation in this forum. Keep exploring, asking questions, and sharing your insights with the community. Together, we can find the best solutions. For clear and precise answers, choose IDNLearn.com. Thanks for stopping by, and come back soon for more valuable insights.