Please help me simplify this and show full working out. (a) (2x^3 divide 3y^4)^2 multiply (x divide 3y)^-2 (b) a-a^-1 divide a^2 - 1 (c) 3^a+2 times 9^2a+1 divide by 6^2 times 3^5a+3
(2x^3/3y^4)^2 * (x/3y)^-2
4x^6/9y^8 * 9y^2/x^2
4x^4/y^6
(a - 1/a) / (a^2-1)
((a^2-1)/a) / (a^2-1)
1/a
3^(a+2) 9^(2a+1) / 6^2 3^(5a+3)
3^(a+2) 3^(4a+2) / 2^2 3^2 3^(5a+3)
3^(5a+4) / 4*3^(5a+5)
1/12
Of course! Let's simplify each expression step by step and show the full working out.
(a) To simplify (2x^3 ÷ 3y^4)^2 × (x ÷ 3y)^-2:
Step 1: Simplify the first part of the expression, (2x^3 ÷ 3y^4)^2:
To square a fraction, we square the numerator and denominator separately. Thus:
(2x^3 ÷ 3y^4)^2 = (2^2x^6 ÷ 3^2y^8) = (4x^6 ÷ 9y^8)
Step 2: Simplify the second part of the expression, (x ÷ 3y)^-2:
To simplify a negative exponent, we can move the base from numerator to denominator (or vice versa) and change the sign of the exponent. Thus:
(x ÷ 3y)^-2 = (3y ÷ x)^2 = (9y^2 ÷ x^2)
Step 3: Multiply the simplified expressions:
(4x^6 ÷ 9y^8) × (9y^2 ÷ x^2) = (4x^6 × 9y^2) ÷ (9y^8 × x^2)
Simplifying further, we get:
= (36x^6y^2) ÷ (9x^2y^8)
= 4x^4 ÷ y^6
Therefore, the simplified expression is 4x^4 ÷ y^6.
(b) To simplify a-a^-1 ÷ a^2 - 1:
Step 1: Simplify the numerator, a-a^-1:
This can be expressed as (a^1 ÷ a^-1) by using the rule that a^m ÷ a^n = a^(m-n).
Simplifying, we have a^(1 - (-1)), which is a^2.
Step 2: Simplify the denominator, a^2 - 1:
This is already in simplified form.
Step 3: Write the simplified expression:
Finally, we can substitute the simplified numerator and denominator into the original expression:
a-a^-1 ÷ a^2 - 1 = a^2 ÷ (a^2 - 1)
Therefore, the simplified expression is a^2 ÷ (a^2 - 1).
(c) To simplify 3^(a+2) × 9^(2a+1) ÷ 6^2 × 3^(5a+3):
Step 1: Simplify the powers of 3, 9, and 6:
3^(a+2) can be written as 3^a × 3^2 = 3^a × 9
9^(2a+1) can be expressed as (3^2)^(2a+1), which is equal to 3^(2(2a+1)) = 3^(4a+2)
6^2 simplifies to 36
3^(5a+3) is already in simplified form.
Step 2: Combine the simplified terms:
(3^a × 9) × (3^(4a+2)) ÷ 36 × 3^(5a+3)
Rearranging the terms, we get:
= (3^a × 3^(4a+2) × 9) ÷ (36 × 3^(5a+3))
Step 3: Use the property of exponents to simplify further:
3^(a + (4a+2)) = 3^(5a+2)
9 can be simplified to 3^2
36 can be simplified to 3^2 × 2^2 = 9 × 4
Therefore, the expression becomes:
= (3^a × 3^(4a+2) × 3^2) ÷ (9 × 4 × 3^(5a+3))
Step 4: Simplify the remaining powers of 3:
Using the property of exponents, we add the exponents when raising a power to another power. Thus:
3^a × 3^(4a+2) × 3^2 = 3^(a + (4a+2) + 2) = 3^(5a+4)
Simplifying the denominator, we have:
9 × 4 × 3^(5a+3) = 36 × 3^(5a+3)
Step 5: Write the simplified expression:
Replacing the simplified forms, we get:
= 3^(5a+4) ÷ (36 × 3^(5a+3))
Therefore, the simplified expression is 3^(5a+4) ÷ (36 × 3^(5a+3)).