(ii) $$ \frac{81^{a x} imes 27^{x}}{9^{x}}=729 $$ (iii) Find the value of $$ x $$ and $$ y $$. $$ 9^{a y} imes 2^{x}=72 $$ (iv) $$ 3^{2 y}=81 $$ Evaluate the following index expressions (i) $$ 25^{3 / 2} imes 9^{3 / 3 / 2} $$ in

Question

(ii) $$ \frac{81^{a x} 	imes 27^{x}}{9^{x}}=729 $$ (iii) Find the value of $$ x $$ and $$ y $$. $$ 9^{a y} 	imes 2^{x}=72 $$ (iv) $$ 3^{2 y}=81 $$ Evaluate the following index expressions (i) $$ 25^{3 / 2} 	imes 9^{3 / 3 / 2} $$ in

UpStudy AI · Accepted Answer

Let's solve the problems step by step.

### Problem (ii)
We have the equation:
$$
\frac{81^{a x} \times 27^{x}}{9^{x}}=729
$$

First, we can express all the numbers in terms of base 3:
- $$ 81 = 3^4 $$
- $$ 27 = 3^3 $$
- $$ 9 = 3^2 $$
- $$ 729 = 3^6 $$

Now, substituting these into the equation gives:
$$
\frac{(3^4)^{a x} \times (3^3)^{x}}{(3^2)^{x}} = 3^6
$$

This simplifies to:
$$
\frac{3^{4ax} \times 3^{3x}}{3^{2x}} = 3^6
$$

Combining the exponents in the numerator:
$$
\frac{3^{4ax + 3x}}{3^{2x}} = 3^6
$$

Now, we can simplify the left side:
$$
3^{4ax + 3x - 2x} = 3^6
$$
$$
3^{4ax + x} = 3^6
$$

Since the bases are the same, we can set the exponents equal to each other:
$$
4ax + x = 6
$$
Factoring out $$ x $$:
$$
x(4a + 1) = 6
$$
Thus, we have:
$$
x = \frac{6}{4a + 1}
$$

### Problem (iii)
We need to solve the equation:
$$
9^{a y} \times 2^{x} = 72
$$

Again, express $$ 9 $$ and $$ 72 $$ in terms of base 3:
- $$ 9 = 3^2 $$
- $$ 72 = 8 \times 9 = 2^3 \times 3^2 $$

Substituting gives:
$$
(3^2)^{a y} \times 2^{x} = 2^3 \times 3^2
$$

This simplifies to:
$$
3^{2ay} \times 2^{x} = 2^3 \times 3^2
$$

Now, we can equate the bases:
1. For base 3:
   $$
   2ay = 2 \implies ay = 1 \implies y = \frac{1}{a}
   $$
2. For base 2:
   $$
   x = 3
   $$

### Problem (iv)
We have the equation:
$$
3^{2y} = 81
$$
Expressing $$ 81 $$ in terms of base 3:
$$
3^{2y} = 3^4
$$

Setting the exponents equal gives:
$$
2y = 4 \implies y = 2
$$

### Summary of Results
From the above calculations:
- $$ x = 3 $$
- $$ y = 2 $$
- $$ a = \frac{1}{2} $$ (from $$ ay = 1 $$)

### Problem (i)
Now, we evaluate the expression:
$$
25^{3/2} \times 9^{3/(3/2)}
$$

First, simplify $$ 9^{3/(3/2)} $$:
$$
9^{3/(3/2)} = 9^{3 \cdot \frac{2}{3}} = 9^2 = 81
$$

Now, simplify $$ 25^{3/2} $$:
$$
25^{3/2} = (5^2)^{3/2} = 5^{2 \cdot \frac{3}{2}} = 5^3 = 125
$$

Now, multiply the two results:
$$
125 \times 81
$$

Calculating $$ 125 \times 81 $$:
$$
125 \times 81 = 10125
$$

Thus, the final result for the index expression is:
$$
\boxed{10125}
$$
$$ x = 3 $$, $$ y = 2 $$, and the value of the expression $$ 25^{3/2} \times 9^{3/(3/2)} $$ is 10125.

(ii) \( \frac{81^{a x} \times 27^{x}}{9^{x}}=729 \) (iii) Find the value of \( x \) and \( y \). \( 9^{a y} \times 2^{x}=72 \) (iv) \( 3^{2 y}=81 \) Evaluate the following index expressions (i) \( 25^{3 / 2} \times 9^{3 / 3 / 2} \) in

Upstudy AI Solution

Answer

Solution

Mind Expander

Related Questions

Latest Algebra Questions