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# Surds & Indices

1.

$$125 \times 125 \times 125 \times 125 \times 125 = 5^?$$

$$125 \times 125 \times 125 \times 125 \times 125 = (5^3 \times 5^3 \times 5^3 \times 5^3 \times 5^3 \times ) = 5^{(3+3+3+3+3+)} = 5^{15}$$

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2.

If $$a^x = b, b^y = c$$ and $$c^z = a$$, then the value of $$xyz$$ is:

$$a^1 = c^z = (b^y)^z = b^{yz} = (a^x)^{yx} = a^{xyz}.$$

Therefore, $$xyz = 1.$$

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3.

If $$2^{(x-y)} = 8$$ and $$2^{(x+y)} = 32$$, then $$x$$ is equal to:

$$2^{(x-y)} = 8 = 2^3$$

$$\Rightarrow x-y = 3 ....(1)$$

$$2^{(x+y)} = 32 = 2^5$$

$$\Rightarrow x+y = 5 ....(2)$$

On solving (1) & (2), we get $$x = 4.$$

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4.

If $$3^x - 3^{x-1} = 18$$, then the value of $$x^x$$ is:

$$3^x - 3^{x-1} = 18$$

$$\Rightarrow 3^{x-1} \times (3-1) = 18$$

$$\Rightarrow 3^{x-1} = 9 = 3^2$$

$$\Rightarrow x-1 = 2$$

$$\Rightarrow x = 3$$.

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5.

If $$7^a = 16807$$, then the value of $$7^{(a-3)}$$ is:

$$7^a = 16807,$$

$$\Rightarrow 7^a = 7^5, a = 5$$

Therefore, $$7^{(a-3)} = 7^{(5-3)} = 7^2 = 49$$