Test: Laws of Exponents - Grade 9 MCQ

(4/5)^-9 ÷ (4/5)^-9 simplifies as follows:
Using the division rule of exponents, subtract the exponents:
(4/5)^-9 ÷ (4/5)^-9 = (4/5)^-9 - (-9) = (4/5)⁰
Any non-zero number raised to the power of 0 is 1:
(4/5)⁰ = 1
The answer is C: 1.

(64)^-3/2 = ( 8 )^{2 x - 3/2} =  (8)^-3  

= 1 / (8)³ = 1/ 512 

So option B is correct answer . 

For finding the value of x, we first need to make RHS in the same format as LHS so that we can easily compare both.
first factorise 45 = 3² x 5¹
[3^(x-3)] 5^(x-4) = 45 = 3² (5¹)
On comparing both sides we get,
x-3 = 2 and x-4 = 1
By solving we get
=> x = 5
Hence, option A is the correct answer.

To solve this problem, we need to evaluate the expression (100 - 99⁰) x 100. Let's break it down step by step:
Step 1: Evaluate 99⁰: Any number raised to the power of 0 is equal to 1. So, 99⁰ = 1.
Step 2: Substitute the value of 99⁰ into the expression: (100 - 1) x 100 = 99 x 100.
Step 3: Multiply 99 and 100: 99 x 100 = 9900.
Therefore, the value of the expression (100 - 99⁰) x 100 is 9900.
Hence, the correct answer is C: 9900.

(25)^1/3=(5²)^1/3=(5)^2/3
5^2/3x5^1/3=5^3/3  = 5^2/3+1/3   = 5¹   [a^m x a ⁿ = a^(m+n)  ]
so answer is 5

Interpret the exponent:

• The fractional exponent 3/2 means:
  16^(3/2) = (16^(1/2))³
  Here, 16^(1/2) is the square root of 16.

Calculate the square root:
16^(1/2) = √16 = 4

Raise the result to the power of 3:
(16^(1/2))³ = 4³ = 64

Using the quotient rule for exponents, (a^m / aⁿ) = a^{(m - n)}:

13^1/5 / 13^1/3 = 13^{(1/5 - 1/3)}
= 13^{1/5 - 1/3}
Taking LCM of 5 and 3, we get 15
Now for 1/5 - 1/3, we get 15 as denominator and (3 - 5) as numerator
= 13 ^{3 - 5 / 15}
= 13^-2/15

Correct answer is D. 16.
= (16^(3/2)) ÷ (16^(1/2)) = ?
= (16)^{3/2  -  1/2}
= (16)¹ 
= 16.

Recognize that 8 can be expressed as 2³, as 2 x 2 x 2 = 8.
Rewrite 8^-5/3 as (2³)^-5/3
Apply the power of a power rule, which states that (a^m)ⁿ = a^mn. In this case,
multiply the exponents: 3 x -5/3
Simplify the expression.
Let's go through the steps:
(2³)^-5/3
(2)^-5
Now, 2^-5 means taking the reciprocal of 2⁵ (since a negative exponent indicates
reciprocal), so: 
2^-5 = 1/  2⁵
= 1/32  [ 2⁵ = 2x2x2x2x2 = 32 ]

Step 1: Simplify ( -3/4 )⁻³
We apply the rule: a⁻ⁿ = 1 / aⁿ

So:
( -3/4 )⁻³ = 1 / ( -3/4 )³

First calculate ( -3/4 )³:
( -3/4 ) × ( -3/4 ) × ( -3/4 ) = (-3)³ / 4³ = -27 / 64

Now,
1 ÷ ( -27 / 64 ) = -64 / 27

Step 2: Check Each Option

• Option a: ( -4/3 )³
  (-4)³ / 3³ = -64 / 27 (matches)

• Option b: ( 3/4 )⁻³
  1 / ( 3/4 )³ = 4³ / 3³ = 64 / 27 (positive, not matching)

• Option c: - ( 3/4 )³

  • ( 27 / 64 ) = -27 / 64 (different)

• Option d: ( 4/3 )³
  64 / 27 (positive, not matching)

Final Answer:
a) ( -4/3 )³

First, evaluate 16^−1/4:

​

Now evaluate

Multiply the two results:

Anything to the power 0 is always 1. 
The answer is 1 because of a⁰= 1.
So 1×1/1 = 1

256^3/4 = ?

The solution to 256^3/4 is calculated as follows:

• 256 can be rewritten as (4⁴).
• Applying the exponent rule, we express this as: (4⁴)^3/4.
• This simplifies to: 4^{(4 * 3/4)} = 4³.
• Calculating 4³ gives us: 64.

To find the value of (-1)^-1, we use the property of exponents that states a^-n = 1/(aⁿ). Applying this property:

• (-1)^-1 = 1/((-1)¹)

Since (-1)¹ is -1, this becomes:

• 1 / (-1) = -1

Therefore, the value of (-1)^-1 is -1.

81^1/2 can be expressed as:

• 81 can be rewritten as (9)^2.
• Thus, 81^1/2 becomes (9)^{2 x 1/2}.
• This simplifies to (9)¹, which equals 9.

The final answer is 9.

To find the value of 49^1/2, we recognise that taking a number to the power of 1/2 is equivalent to finding its square root. Thus, we can express this as:

• 49^1/2 = √49
• √49 = 7

 ( 5 ) ^{3 x -1/3}   x  ( 5 ) ^{2  x -1/2} 

  ( 5 ) ^-1  x   ( 5 )  ^-1

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- The expression  (a^m)ⁿ involves exponentiation rules.
- The rule for powers of a power is (a^m)ⁿ = a^mn
- This means you multiply the exponents when raising a power to another power.

- Very small numbers in standard form are expressed using negative exponents.
- Standard form is a way of writing numbers as a product of a number between 1 and 10, and a power of 10.
- For small numbers, the power of 10 is negative, indicating division by a power of 10.
- Example: 0.001 is written as 10^-3
- Negative exponents simplify representing very small values efficiently.