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UPTET Paper 2 Math and Science 2019 Solved Paper

Question : 92 of 120

Marks: +1, -0

(\frac{9}{7})^{3}\times (\frac{49}{81})^{2x-6}=(\frac{7}{9})^{9}

, then the value of x is

Solution:

Concept:

Use the laws of exponents:

(a^m)^n = a^{mn}

and

a^m \times a^n = a^{m+n}

. Also,

\left(\frac{9}{7}\right)^3 = \left(\frac{7}{9}\right)^{-3}

because the reciprocal changes the sign of the exponent.

Explanation:

Rewrite

\frac{49}{81}

\left(\frac{7}{9}\right)^2

. Then

\left(\frac{49}{81}\right)^{2x-6} = \left(\frac{7}{9}\right)^{2(2x-6)} = \left(\frac{7}{9}\right)^{4x-12}

Now the left side is

\left(\frac{9}{7}\right)^3 \times \left(\frac{7}{9}\right)^{4x-12}

. Write

\left(\frac{9}{7}\right)^3 = \left(\frac{7}{9}\right)^{-3}

So the left side becomes

\left(\frac{7}{9}\right)^{-3} \times \left(\frac{7}{9}\right)^{4x-12} = \left(\frac{7}{9}\right)^{4x-15}

Set this equal to the right side

\left(\frac{7}{9}\right)^9

. Since the bases are the same, equate the exponents:

4x - 15 = 9

Solve:

4x = 24

, so

x = 6

Answer:

x = 6

(Option C)

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