4) Simplify :
(i) (125 ×x^(-3))/(5^(-3)×25 ×x^(-6) )
(ii) (16×10^2×64)/(2^4×4^2 )
it's better to use * for multiplication, so it does not get confused with x, a variable. So,
(125*x^(-3))/(5^(-3)*25*x^(-6) )
= (5^3 x^-3)/(5^-3 * 5^2 x^-6)
= (5^3 x^-3)/(5*-1 x^-6)
= 5^(3-(-1)) x^(-3-(-6))
= 5^4 x^3
= 625 x^3
(i) (125 ×x^(-3))/(5^(-3)×25 ×x^(-6) )
= [5*5*5/x*x*x ] / [ (5*5)/ {5*5*5} / (x*x*x*x*x*x) ]
= [5*5*5/x*x*x ] * [ {5*5*5} / (x*x*x*x*x*x) ] / (5*5)
= 5*5*5*5 * (x*x*x)
= 625 x^3
your turn now
hint
16 = 4^2
64 = 16* 4 = 4^3
2^4 =( 2^2)^2 = 4^2
(125 ×x^(-3))/(5^(-3)×25 ×x^(-6) )
write it this way, easier to read:
(125x^-3)/(5^-3*25*x^-6 ) , the * is often used as multiplication, if an exponent is a monomial you don't need brackets around it.
= (125/x^3)/((1/125)(25))*x^6
= 125*5*x^3
= 625x^3
or, even simpler ...
(125x^-3)/(5^-3*25*x^-6 )
= 125/(25/125) x^(-3+6)
= 625x^3 or (5^4)(x3)
looking at the 2nd one, I see all powers of 2 and a 10^2
(16×10^2×64)/(2^4×4^2 )
= (2^4)(100)(2^6)/( (2^4)(*2^4))
= 100(2^10)/2^8
= 100(2^2) = ...
To simplify the given expressions, let's break down the steps:
(i) Simplify: (125 ×x^(-3))/(5^(-3)×25 ×x^(-6))
Step 1: Let's simplify the terms with exponents first.
- For the numerator, x^(-3) means 1/x^3.
- For the denominator, x^(-6) means 1/x^6.
Now the expression becomes:
(125 × 1/x^3) / (5^(-3) × 25 × 1/x^6)
Step 2: Simplify the exponents using the rule:
a^(-n) = 1 / (a^n)
The expression becomes:
(125 × 1/x^3) / (1 / (5^3 × 25 × x^6))
Simplifying further:
(125 × 1/x^3) / (1 / (125 × 25 × x^6))
Step 3: Invert the denominator and multiply:
(125 × 1/x^3) × (125 × 25 × x^6)
Step 4: Simplify the expression:
125 × 125 × 25 × x^6 / x^3
= 125^2 × 25 × x^(6-3)
= 125^2 × 25 × x^3
= 156250 x^3
Therefore, the simplified expression is 156250 x^3.
(ii) Simplify: (16×10^2×64)/(2^4×4^2)
Step 1: Let's simplify the terms with exponents first.
For the denominator, 2^4 means 2 × 2 × 2 × 2 = 16.
For the denominator, 4^2 means 4 × 4 = 16.
Now the expression becomes:
(16 × 10^2 × 64) / (16 × 16)
Step 2: Simplify the expression:
Cancelling out the common factors:
(10^2 × 64) / 1
Simplifying further:
(100 × 64) / 1
= 6400
Therefore, the simplified expression is 6400.