Practice Questions

Recall the law of exponents used to simplify an expression where powers with different bases but the same exponent are multiplied, such as aᵐ × bᵐ.

Calculate the multiplicative inverse of (-6)^(-3).

Critique the statement: 'A number with a negative exponent is always a negative number.' Justify your critique with at least two different examples.

Recall the law of exponents that applies to an expression like (aᵐ)ⁿ.

Calculate the value of (1/5)^(-3).

Identify the base and the exponent in the expression (-7)⁻³.

Apply the laws of exponents to express (27)^(-5) as a power with base 3.

Recall the value of any non-zero integer 'a' raised to the power of zero.

Formulate an expression involving at least three different laws of exponents that simplifies to x^(-1).

Solve for the value of the expression (2^(-1) + 3^(-1) + 4^(-1)) * (24)^0.

Name the form used to conveniently write very large or very small numbers, such as 5.97 x 10²⁴ kg.

Evaluate whether standard form is a more efficient way to represent and use the mass of a proton, approximately 0.00000000000000000000000167 kg, compared to its usual form. Justify your reasoning.

Explain the law for multiplying two powers with the same base, such as 5³ × 5⁻⁷.

Examine the equation 8 * 2^(n+2) = 32 and solve for n.

A scientist claims that Virus A, with a diameter of 2.4 × 10^(-7) m, is approximately half the size of Bacterium B, with a diameter of 4.8 × 10^(-7) m. Evaluate this claim and justify your conclusion.

Design a real-world problem where an engineer must calculate the total thickness of a composite material. The material is made of 50 layers of a metal foil, each 5 × 10^(-5) m thick, and 50 layers of a polymer sheet, each 1.5 × 10^(-4) m thick. Propose a method to calculate the total thickness and express the final answer in standard form.

Analyze and calculate the value of the expression (5^0 + 5^(-1)) / (5^(-1) - 5^0).

Compare the mass of Planet A, which is 6.42 x 10^23 kg, with the mass of Planet B, which is 1.898 x 10^27 kg.

Solve this problem: The size of a particular virus is 1.5 x 10^(-6) m and the size of a dust particle is 7.5 x 10^(-5) m. Calculate how many times larger the dust particle is compared to the virus.

Calculate the value of (2)^(-4) * (3)^(-4) * (4)^(-4).

Explain the relationship between a number with a negative exponent, like a⁻ᵐ, and its equivalent with a positive exponent.

Explain what the expression (a/b)ᵐ is equal to, according to the laws of exponents.

Solve for the value of 'x' in the equation: (7/4)^(-3) * (7/4)^(-5) = (7/4)^(x-2).

Recall the simplified form of the expression 1 / a⁻ᵐ.

Define the term 'multiplicative inverse' in the context of exponents.

List two reasons why it is useful to express numbers in standard form.

Describe the steps to express a very small number, such as 0.000047, in standard form.

Propose a method to determine which is larger, (2^(-3))^4 or (2^(-4))^3, without calculating their final numerical values. Justify your method.

Create a word problem to compare the volumes of Planet X and Planet Y. The radius of Planet X is 6 × 10^6 m, and the radius of Planet Y is 3 × 10^7 m. Evaluate the ratio of the volume of Planet Y to Planet X, assuming both are perfect spheres. (Volume of a sphere = (4/3)πr^3).

Apply the laws of exponents to simplify (x^(-2) * y^3)^(-3) and express the result with positive exponents.

Formulate a general rule to find the multiplicative inverse of (a/b)^(-n), where 'a' and 'b' are non-zero integers and 'n' is a positive integer. Justify your formula using the laws of exponents.

Justify why any non-zero number 'a' raised to the power of zero is equal to 1. Use the law of exponents a^m ÷ a^n = a^(m-n) in your justification.

A student simplified (3^(-1) + 4^(-1))^(-1) to 3 + 4 = 7. Critique this method, explain why it is incorrect, and propose the correct method to evaluate the expression.

Justify the statement: 'The product of a number a^m and its multiplicative inverse is always 1, for any non-zero 'a' and any integer 'm'.'

Describe how to write the number 1256.249 in expanded form using powers of 10.

Solve the following: {(1/4)^(-3) - (1/5)^(-2)} / (1/7)^(-2).

Summarize how the pattern of dividing by the base leads from positive exponents to negative exponents, using 3 as the base.

List the five main laws of exponents that apply to any non-zero integers 'a' and 'b', and any integer exponents 'm' and 'n'.

Evaluate the expression [ (x^(-2) * y^3) / (x^3 * y^(-2)) ]^(-1) and justify each step of the simplification process using the appropriate laws of exponents.

The distance from Earth to Star A is 4.37 x 10^16 meters, and the distance from Earth to Star B is 8.5 x 10^15 meters. Calculate the difference between these two distances and express the result in standard form.

Create an equation to find the value of 'm' where the simplification of the expression equals (5/3)^9: (27/125)^(2m-1) × (9/25)^(-3). Formulate the steps to solve for 'm' and justify each step using the laws of exponents.

Analyze and simplify the expression [(2/3)^(-2)]^3 * (1/3)^(-4) * 3^(-1) * (1/6).

Demonstrate the simplification of the expression (36 * x^(-4)) / (6^(-3) * 12 * x^(-8)) where x is not zero. Present the final answer using only positive exponents.

Design an expression using variables p and q, involving negative exponents and at least two laws of exponents, such that when p=2 and q=3, the value of the expression is 1/72. Justify that your design works.

Create an equation of the form (a/b)^x × (a/b)^y = (b/a)^z. Formulate a solution to express 'z' in terms of 'x' and 'y', justifying your steps.