Properties of Exponents

Write the following repeated multiplications as a power and then calculate their numerical value. Remember that the base is the number that is 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 as 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 the product of 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 property of the quotient (division) of powers with the same base to reduce the following expressions to a single power. Remember that you must 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 result of the following expression, applying the properties of powers before calculating the final value. Note carefully that all the bases are the same negative number:

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

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

Calculate the value of the following powers taking into account the properties of the power with exponent 0 and the power with 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) $123^0 + 1^{20}$

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

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

Hint: Notice that the last fraction has its base inverted with respect to the first two.

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

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

Simplify the following expressions using the property of the power of a power. You must leave the result expressed 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 the same. 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 $3^4$ inches.

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.

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

Hint: Transform all bases to base 3 before operating on the exponents.

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 that allows you to compare them mentally.

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

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

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

1. We start with a solid square of side 1 foot (Iteration 0).

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

3. We 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.

A giant Rubik's cube is made up 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 as a power of base 3.

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

An external hard drive has a storage capacity of 1 TB (Terabyte). Knowing that the approximate equivalents 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 on the hard drive in total? Express the result as a power of base 2.

A cloud data storage server has a total capacity of 16 Petabytes (PB). Knowing that computing equivalents 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 on the server in total? Express the result as a power of base 2.

Calculate the exact value of the following combined operation, first applying the properties of powers and transforming to standard form before operating.

$$ \frac{ 3,5 \cdot 10^{-5} + 150 \cdot 10^{-7} }{ 0,02 \cdot 10^4 \cdot (10^{-2})^3 } $$

Express the final result in standard form.