1.1.5 Understanding and calculating with powers and roots

• The square root of 16 is 4 because $4 \times 4 =$$16$.


  True
• When calculating $3^{ - 2}$, the exponent becomes positive after flipping the base


  True
• What is the formula for multiplying powers with the same base?

  $a^{m} \times a^{n} =$$a^{m + n}$
• What is the result of $\frac{5^{4}}{5^{2}}$?


  $5^{2}$
• When dividing powers with the same base, you subtract their exponents.

  True
• Match the property of powers with its formula:

  Multiplying powers with same base ↔️ $a^{m} \times a^{n} =$$a^{m + n}$

  Dividing powers with same base ↔️ $\frac{a^{m}}{a^{n}} =$$a^{m - n}$

  Power of a power ↔️ $(a^{m})^{n} =$$a^{m \times n}$
• What is the result of $2^{2} \times 2^{3}$?


  $2^{5}$
• What does the term "higher order roots" extend from?

  Square and cube roots
• Match the negative power with its calculated value:

  $3^{ - 2}$ ↔️ $\frac{1}{9}$

  $2^{ - 3}$ ↔️ $\frac{1}{8}$

  $5^{ - 1}$ ↔️ $\frac{1}{5}$

  $4^{ - 2}$ ↔️ $\frac{1}{16}$
• What does $2^{3}$ mean?


  $2 \times 2 \times 2$
• Match the root type with its definition:

  Square Root ↔️ Number multiplied by itself

  Cube Root ↔️ Number multiplied by itself three times
• Steps to calculate with negative powers

  1️⃣ Flip the base to its reciprocal

  2️⃣ Change the exponent to positive

  3️⃣ Apply the positive exponent to the reciprocal base
• What is the result of $(3^{2})^{3}$?


  $3^{6}$
• $3^{4}$ equals 81 because we multiply 3 by itself 4 times.


  True
• What is the value of $2^{ - 3}$?


  $\frac{1}{8}$
• The property "Power of a power" states that $(a^{m})^{n} =$$a^{m \times n}$.


  True
• What does a negative exponent indicate?

  Reciprocal operation
• Match the term with its value in the example $2^{ - 3}$:


  Base ↔️ 2

  Exponent ↔️ -3

  Result ↔️ $\frac{1}{8}$
• A negative power means you are sharing a piece of a cookie.

  True
• Match the root type with its example:

  Square Root ↔️ $\sqrt{16}$

  Cube Root ↔️ $\sqrt[3]{8}$
• Match the radical form with its corresponding power form:

  $\sqrt[n]{a}$ ↔️ $a^{\frac{1}{n}}$

  $\sqrt[n]{a^{m}}$ ↔️ $a^{\frac{m}{n}}$
• The cube root of 8 is 2 because $2^{3} =$$8$


  True
• Steps to calculate a root:

  1️⃣ Identify the original number

  2️⃣ Determine the root type (e.g., square root, cube root)

  3️⃣ Find a number that, when multiplied by itself the required number of times, equals the original number

  4️⃣ Write the result as the root
• The root of a radical becomes the bottom number of a fraction when converting to powers.

  True
• Steps to convert a radical to a power:

  1️⃣ Identify the root of the radical

  2️⃣ Determine the exponent inside the root

  3️⃣ Write the root as the bottom number of the fraction

  4️⃣ Write the exponent as the top number of the fraction
• The square root of 9 is 3 because $3 \times 3 =$$9$


  True
• Match the root type with its definition:

  Square Root ↔️ Find $x$ such that $x^{2} =$$a$

  Cube Root ↔️ Find $x$ such that $x^{3} =$$a$

  Higher Order Roots ↔️ Find $x$ such that $x^{n} =$$a$
• What is the square root of 25?

  5
• What are roots often compared to in the analogy of plant growth?

  Seeds
• What operation is used to calculate square roots?

  Multiplication
• Why is the 4th root of 16 equal to 2?

  $2^{4} =$$16$
• What does a negative power indicate about sharing?

  Sharing into tiny pieces
• Match the radical expression with its equivalent power:

  $\sqrt[3]{5^{2}}$ ↔️ $5^{\frac{2}{3}}$

  $\sqrt[4]{7^{3}}$ ↔️ $7^{\frac{3}{4}}$

  $\sqrt[5]{2^{4}}$ ↔️ $2^{\frac{4}{5}}$

  $\sqrt[2]{3^{1}}$ ↔️ $3^{\frac{1}{2}}$
• Match the number with its square root:

  9 ↔️ 3

  16 ↔️ 4

  25 ↔️ 5
• The square root of 16 is 4 because 4 multiplied by itself equals 16

  True
• What symbol is used to represent square roots?

  $\sqrt{ }$
• What does the index of a radical become in its power form?

  Denominator