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

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:

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

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) $\frac{a^6}{a}$

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

We have a square whose side measures $2^4$ centimeters.

1. Express the perimeter of the square as a single power of base 2 (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 2.

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:

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

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.