Guided Practice

When you multiply two powers with the same base, you add the exponents: 3^{7} • 3^{9} = 3^{16}.

When you divide two powers with the same base, you subtract the exponents: 3^{7} ÷ 3^{9} = 3^{–2} = 1/9.

When you raise a power to a power, you multiply the exponents: (3^{7})^{9} = 3^{63}.

Write each power expression as an integer or decimal.

6^{2} • 6^{3} =

8^{5} ÷ 8^{7} =

(2^{3})^{2} =

In an expression like 2^{3}, the number 2 is called the base, and 3 is called the exponent. The entire expression is called a power.

When the exponent is a positive whole number, it tells how many times the base is used as a factor.

3^{5} = 3 × 3 × 3 × 3 × 3

Any power with an exponent of 0 is equal to 1

(a^{0} = 1)—except for 0^{0}, which is undefined. Any power with an exponent of 1 is equal to the base: 10^{1} = 10.

You can multiply powers. The expression 8 × 8 × 8 can be written as 8^{3}. But, since 8 = 2^{3}, the expression 8 × 8 × 8 can also be written as 2^{3} × 2^{3} × 2^{3}. This is the same as a multiplication expression with nine 2s:

So, 2^{3} × 2^{3} × 2^{3} = 2^{9}.

When you multiply powers with the same base, you add the exponents: a^{m} × a^{n} = a^{m + n}.

A number written without an exponent has an exponent of 1: 84 = 84^{1}.

You can divide powers too. You know that 9 ÷ 3 = 3. This is the same as 3^{2} ÷ 3^{1} = 3^{1}.

Or, 1000 ÷ 100 = 10. This is the same as 10^{3} ÷ 10^{2} = 10^{1}.

Notice the pattern?

3^{2} ÷ 3^{1} = 3^{1} 10^{3} ÷ 10^{2} = 10^{1}

When you divide powers with the same base, you subtract the exponents: a^{m} ÷ a^{n} = a^{m – n}.

A chess board has squares.

There were , or 2 × 2, grains of rice on the third square.

There were , or 2 × 2 × 2, grains of rice on the fourth square.

There would be , or 16, grains of rice on the fifth square.

There was , or 1, grain of rice on the first square.

Write the power as an integer.

3^{5} =

Write the expression as a power with a base of 8 and as an integer.

8 × 8 × 8

Power:

Integer:

Write the expression as a power with a base

of 5 and as an integer. (Use commas in integers greater than 999.)

5 × 5 × 5 × 5 × 5 × 5 × 5

Power:

Integer:

Write the expression as a power with a base

of 12 and as an integer. (Use commas in integers greater than 999.)

12 × 12 × 12 × 12 × 12 × 12

Power:

Integer:

Write the expression as a power with a base

of 15.

1

Power:

Write the expression as a power with a base of 4.

4

Power:

Write the expression as a power with a base of 2 and as an integer.

8 × 8 × 8

Power:

Integer:

Write the product as a power with a base

of 6 and as an integer. (Use commas in integers greater than 999.)

6^{2} × 6^{5}

Power:

Integer:

Write the product as a power with a base

of 7 and as an integer. (Use commas in integers greater than 999.)

7^{4} × 7^{2} × 7^{3}

Power:

Integer:

Write the product as a power with a base

of 9 and as an integer. (Use commas in integers greater than 999.)

9^{5} × 9 × 9^{2} × 9^{2}

Power:

Integer:

Write the product as a power with a base

of 13 and as an integer. (Use commas in integers greater than 999.)

13 × 13^{2} × 1

Power:

Integer:

Write the quotient as a power with a base

of 2 and as an integer.

2^{9} ÷ 2^{6}

Power:

Integer:

Write the quotient as a power with a base

of 10 and as an integer.

10^{6} ÷ 10^{5}

Power:

Integer:

Write the quotient as a power with a base

of 8 and as an integer.

8^{2} ÷ 8

Power:

Integer:

Write the quotient as a power with a base

of 12 and as an integer.

12^{2} ÷ 1

Power:

Integer:

Exponents can be negative numbers. For example, if we divide 2^{3} by 2^{4}, we get 2^{3 – 4}, or 2^{–1}. And 2^{3} ÷ 2^{4} = 8 ÷ 16, or 1/2. So, 2^{–1} = 1/2.

The product 2^{–1} × 2^{2} is equal to 2^{–1 + 2}, or 2^{1}. This is the same as 1/2 × 4 = 2.

In general, a^{–m} = 1/a^{m}.

A power with a positive base can never be less than 0, even if the exponent is **really** negative.

Remember that ratios indicate division.

The ratio above can be written as a complex fraction:

So, when you have a division expression where both terms have negative exponents, you can rewrite it like this: a^{–m} ÷ a^{–n} = a^{n} ÷ a^{m}, or:

a^{–m}/a^{–n} = a^{n}/a^{m}.

Write the quotient as a power with a base of 2 and as a fraction in simplest form (use a slash: /).

2^{4} ÷ 2^{5}

Power:

Fraction:

Write the quotient as a power with a base of 12 and as a fraction in simplest form (use a slash: /).

12 ÷ 12^{3}

Power:

Fraction:

Write the product as a power with a base of 10 and as a fraction in simplest form (use a slash: /).

10^{–1} × 10^{–1}

Power:

Fraction:

Write the product as a power with a base of 3 and as a fraction in simplest form (use a slash: /).

3^{–5} × 3^{2}

Power:

Fraction:

Write the quotient as a power with a base of 5 and as a fraction in simplest form (use a slash: /).

5^{–3} ÷ 5^{–1}

Power:

Fraction:

Rewrite the expression as a ratio of integers in simplest form.

6^{–2}/6^{–1} = /

Rewrite the expression as a ratio of integers in simplest form.

9^{–2}/9 = /

Rewrite the expression as a ratio of integers in simplest form.

12^{–2}/12^{–3} = /

Rewrite the expression as a ratio of integers in simplest form.

8^{–4}/8^{–6} = /

When you raise a power to a power, you multiply the exponents.

(2^{3})^{2} = 2^{3 × 2} = 2^{6}

Notice that you can switch the order of the exponents, and the power is the same:

(2^{3})^{2} = (2^{2})^{3}, or 8^{2} = 4^{3}

And notice how the power-to-a-power rule works with an exponent of 0:

(2^{3})^{0} = (2^{0})^{3}, or 8^{0} = 1^{3}

Write the expression as a power with a base of 6 and just one exponent. Write the expression also as an integer. Use commas in numbers greater than 999.

(6^{2})^{3}

Power:

Integer:

Write the expression as a power with a base of 11 and just one exponent. Write the expression also as a fraction. Use a slash (/) for fractions.

(11^{2})^{–1} × 11

Power:

Fraction:

Write the expression as a power with a base of

–5 and just one exponent. Write the expression also as an integer.

((–5)^{3})^{4} ÷ (–5)^{10}

Power:

()Integer:

Write the expression as a power with a base of 14 and just one exponent. Write the expression also as an integer.

(14^{0})^{–3} ÷ 14^{–1}

Power:

Integer:

Write the expression as a power with a base of 8 and just one exponent. Write the expression also as an integer.

(8^{–2})^{3} × 8 × 8^{5}

Power:

Integer:

The crowd waits silently—almost silently—in the dark hall while the orchestra begins its warmup. One of the oboists plays a single note, an A note, three times. The rest of the orchestra then uses this note to tune their instruments.

Let's warm up with the orchestra. Press the Warm-Up button below. Then press the orange squares or the keys on the keyboard to hear if the orchestra is in tune.

The tuning note played by the oboe, A, is a sound wave that vibrates at a certain frequency. This frequency is given in a special unit called Hertz. The A note vibrates at 440 Hertz (Hz).

Move both sliders all the way to the right. Now the top left orange button plays A (440) and the top right button plays a higher A (880). (Try them!)

In music, the distance between an A note and the next higher A note (or a B note and the next higher B note, etc.) is called an octave (AHK-tiv). Press the button to play all the octaves.

Try it on the keyboard. When two notes are an octave apart, what do you notice about the approximate ratio of their frequencies in Hertz?

One way to remember what the interval of an octave sounds like is to think of the song "Somewhere Over the Rainbow". Press the button to listen to the song. Then press the karaoke button and try to play along with the song on the keyboard. Practice makes perfect!

Scroll down for more. Let's explore other musical intervals and the ratios of their frequencies . . .

An octave is one kind of interval. Another kind is called a major third. Move both sliders to the fourth tick mark from the left (count the leftmost tick mark as 0). Now the orange squares on the sides of each slider play a major third interval. Press the Major Thirds button to hear some intervals of a major third.

Try it on the keyboard. What is the approximate ratio of the frequencies of the notes in any major third? What is the ratio now of the frequencies directly above and below each other on the sliders?

A way to remember what the interval of a major third sounds like is to think of the first two notes of the melody in the song "When the Saints Go Marching In".

Press the button to listen to the song. Then press it again to play along with the song on the keyboard. What major thirds can you identify in the song? How can you prove that they are major thirds?

Scroll down for more.

Two notes in the interval of an octave are separated by 12 half-steps. Two notes in the interval of a major third are separated by 4 half-steps. And two notes in the interval of a major sixth are separated by 9 half-steps.

The first two notes of the song "My Bonnie Lies Over the Ocean" are in the interval of a major sixth. Press the Major Sixths button to hear some intervals of a major sixth. Then press the song button and play along with the song on the keyboard. How can you play a major sixth using the sliders? What is the ratio of the frequencies?

Here are some of the other musical intervals that musicians know:

minor second: | 1 half-step |

major second: | 2 half-steps |

minor third: | 3 half-steps |

perfect fourth: | 5 half-steps |

perfect fifth: | 7 half-steps |

minor sixth: | 8 half-steps |

Determine their frequency ratios. What patterns do you notice in the frequency ratios and on the sliders?

Can you generate any interval you learned about given any starting frequency? What about given any ending frequency?

Start freq. → ← End freq.

What is the smallest frequency ratio you can detect? What is the highest frequency you can hear? People over the age of 25 commonly have trouble hearing frequencies above 15000 Hertz. Younger people can hear higher frequencies, all the way to about 20000 Hertz.