Question

(i) `(a^x/a^y)^(x + y) * (a^y/a^z)^(y + z) * (a^z/a^x)^(z + x) = 1`

(ii) a^m × aⁿ = a^{m – n}

Options

• Only (i)

• Only (ii)

• Both (i) and (ii)

• Neither (i) nor (ii)

Answer 1

Only (i)

Explanation:

Let’s analyze each statement:

(i) `(a^x/a^y)^(x + y) * (a^y/a^z)^(y + z) * (a^z/a^x)^(z + x) = 1`

Simplify each term using the law of exponents: `a^m/a^n = a^(m - n)`:

`(a^(x - y))^(x + y) * (a^(y - z))^(y + z) * (a^(z - x))^(z + x)`

= `a^((x - y)(x + y)) * a^((y - z)(y + z)) * a^((z - x)(z + x))`

Using power of a power law: a^m × aⁿ = a^{m + n}, combine exponents:

`a^((x - y)(x + y) + (y - z)(y + z) + (z - x)(z + x))`

Calculate each term:

(x – y)(x + y) = x² – y² 

(y – z)(y + z) = y² – z² 

(z – x)(z + x) = z² – x²

Sum of these exponents:

x² – y² + y² – z² + z² – x² = 0

So the exponent sum is 0, therefore

a⁰ = 1

So statement (i) is valid.

(ii) a^m × aⁿ = a^{m – n}

According to the laws of exponents,

a^m × aⁿ = a^{m + n}

Not a^{m – n}.

So statement (ii) is invalid.