Exercises: Operations with powers

Write the following repeated multiplications as a power and then calculate their numerical value. Remember that the base is the number being repeated and the exponent indicates how many times it appears.

1. a) $3 \times 3 \times 3 \times 3$
2. b) $(-2) \times (-2) \times (-2)$
3. c) $10 \times 10 \times 10 \times 10 \times 10$
4. d) $\frac{1}{2} \times \frac{1}{2} \times \frac{1}{2}$

Use the properties of multiplying powers with the same base to simplify these expressions and leave them as a single power. You do not need to calculate the final result, just express the resulting power.

1. a) $5^3 \times 5^4$
2. b) $(-3)^2 \times (-3)^5$
3. c) $x^2 \times x^6 \times x$
4. d) $10^3 \times 10^2 \times 10^4$

Apply the quotient property (division) of powers with the same base to simplify the following expressions to a single power. Remember to subtract the exponents.

1. a) $\frac{7^8}{7^3}$
2. b) $2^{12} : 2^{10}$
3. c) $\frac{(-4)^5}{(-4)^2}$
4. d) $\frac{a^6}{a}$

We have a classic Rubik's cube. We know that on each edge (side) there are 3 small pieces.

Express using a power how many small pieces make up the whole cube and calculate the value.

Look at the drawing to help you:

Calculate the value of the following powers taking into account the properties of zero exponents and powers of base 1. Be careful with the signs.

1. a) $5^0$
2. b) $23^1$
3. c) $1^{15}$
4. d) $(-8)^0$
5. e) $123^0 + 1^{20}$

Simplify the following expressions using the power of a power property. Leave your answer as a single power.

1. a) $(3^2)^4$
2. b) $(x^3)^5$
3. c) $((-2)^2)^3$
4. d) $((a^2)^3)^4$

In these cases, the bases are different but the exponents are equal. Apply the corresponding properties to group the bases and express the result as a single power.

1. a) $2^3 \times 5^3$
2. b) $12^4 : 4^4$
3. c) $(-2)^5 \times 3^5$
4. d) $\frac{20^2}{5^2}$

Fully simplify the following expressions by combining the properties of powers you have learned (product and quotient of equal bases). Express the result as a single power.

1. a) $\frac{2^5 \times 2^3}{2^4}$
2. b) $\frac{x^7}{x^2 \times x^3}$
3. c) $(3^2 \times 3^4) : 3^5$
4. d) $\frac{10^6 \times 10^2}{10^8}$

Work out the result of the following expression, applying the properties of powers before calculating the final value. Note that all bases are the same negative number:

$$ \frac{(-3)^5 \times (-3)^2 \times (-3)}{(-3)^4} $$

Remember that when a number does not have a visible exponent, it is 1.

Simplify the following operation and express the result as a single power of base 2. To do this, you will first need to break down into prime factors any bases that are not prime numbers:

$$ 16 \times 2^5 \times 4^3 $$

We have a square whose side measures $3^4$ centimeters.

1. Express the perimeter of the square as a single power of base 3 (you do not need to calculate the final number, just the power).

2. Express the area of the square as a single power of base 3.

Calculate and simplify the following expression using the properties of powers with rational bases. Express the result as an irreducible fraction with a positive exponent:

$$ \left[ \left( \frac{2}{5} \right)^{-2} \times \left( \frac{2}{5} \right)^5 \right] : \left( \frac{5}{2} \right)^2 $$

An external hard drive has a storage capacity of 1 TB (Terabyte). Knowing that the approximate equivalences in the binary system are:

• $1 \text{ TB} = 2^{10} \text{ GB}$

• $1 \text{ GB} = 2^{10} \text{ MB}$

If a high-quality photograph takes up $2^4$ MB, how many photographs fit in total on the hard drive? Express the result as a power of base 2.

Simplify the following algebraic expression using the properties of powers. The final result must not have negative exponents:

$$ \frac{a^3 \times b^{-4} \times a^{-2}}{a^5 \times b^{-6}} $$

Find the value of the unknown $x$ in each of the following equations, using the properties of powers to equate the exponents:

1. a) $5^x \times 5^3 = 5^9$
2. b) $(3^4)^x = 3^{12}$
3. c) $\frac{7^8}{7^x} = 7^2$

Simplify the following numerical fraction by first factorising the bases into prime factors and then applying the properties of powers:

$$ \frac{15^2 \times 4^2}{6^3 \times 10^2} $$

Simplify the following expression to obtain a single power with a positive prime base. To do this, you will need to apply the properties of powers, factorise the bases into prime factors, and be very careful with signs and negative exponents.

$$ \frac{(-2)^4 \times (2^3)^{-2} \times 16}{\left(\frac{1}{2}\right)^3 \times (-8)^{-2}} $$

Remember that factorising numbers like 16 and 8 into base 2 is the key first step.

Observe the following sequence of figures generated by a fractal process.

1. Start with a solid square of side 1 meter (Iteration 0).

2. Divide the square into 4 equal squares and remove the top right one (Iteration 1).

3. Repeat the process with each of the 3 remaining squares (Iteration 2).

Calculate the total area (the sum of the areas of the black squares) in Iteration 5. Express the result as a single power.

Find the value of the unknown $x$ in the following exponential equation, using the properties of powers to collect terms. All bases can be expressed as powers of the same prime number.

$$ 27^2 \times 9^{-x} = \frac{1}{3^x \times 81} $$

A cloud data storage server has a total capacity of 16 Petabytes (PB). Knowing that computing equivalences are powers of 2 (not 10) according to the following table:

• $1 \text{ KB} = 2^{10} \text{ bytes}$

• $1 \text{ MB} = 2^{10} \text{ KB}$

• $1 \text{ GB} = 2^{10} \text{ MB}$

• $1 \text{ TB} = 2^{10} \text{ GB}$

• $1 \text{ PB} = 2^{10} \text{ TB}$

If a high-resolution photograph file takes up $2^{24}$ bytes, how many photographs fit in total on the server? Express the result as a power of base 2.

Without using a calculator to find the final value, order the following numbers from smallest to largest. To do this, you will need to find a common exponent or a common base to allow you to compare them mentally.

$$ A = 2^{45} \quad ; \quad B = 3^{36} \quad ; \quad C = 5^{18} $$

Help: Try expressing all exponents as multiples of the same number (find the highest common factor of the exponents).

Simplify the following algebraic expression using the properties of powers. The final result must not contain negative exponents or brackets.

$$ \left( \frac{a^{-2} \times b^3}{a \times b^{-1}} \right)^{-3} : \left( \frac{a^3}{b^2} \right)^2 $$

A giant Rubik's cube is made of small unit cubes. We know that the total volume of the large cube is $27^4$ cubic units.

1. Express the length of the edge of the large cube as a power of base 3.

2. If we paint the outer face of each unit cube red, what is the total area painted red on the entire giant cube? Express the result in the form $2 \times 3^n$.

(Remember that the large cube has 6 faces and the area of a face is side $\times$ side).

Work out the exact value of the following combined operation, applying the properties of powers first and converting to standard form before working it out.

$$ \frac{ 3.5 \times 10^{-5} + 150 \times 10^{-7} }{ 0.02 \times 10^4 \times (10^{-2})^3 } $$

Express the final result in standard form.