Algebra Exponents and Expressions Quiz
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Questions and Answers

Which of the following expressions are equal?

  • b³a²
  • a³b² (correct)
  • a²b³
  • b²a³ (correct)

A³b² and a²b³ are the same expression.

False (B)

What is the value of 2⁸?

256

The expression a²b³ can also be written as __________.

<p>ab × ab × ab</p> Signup and view all the answers

Match each expression with its equivalent:

<p>a³b² = b²a³ a²b³ = b³a² b³a² = a²b³ b²a³ = a³b²</p> Signup and view all the answers

Which of the following numbers is equivalent to $10^5$?

<p>100,000 (C)</p> Signup and view all the answers

The number $10^3$ is read as ten raised to the power of three.

<p>True (A)</p> Signup and view all the answers

What is the base in the expression $81 = 3^4$?

<p>3</p> Signup and view all the answers

The mass of Uranus is approximately _____ kg.

<p>86,800,000,000,000,000,000,000,000</p> Signup and view all the answers

Match the following numbers with their exponential forms:

<p>10,000 = $10^4$ 1,000 = $10^3$ 100,000 = $10^5$ 81 = $3^4$</p> Signup and view all the answers

What is the value of $10^2$?

<p>100 (C)</p> Signup and view all the answers

Exponents can only have a base of 10.

<p>False (B)</p> Signup and view all the answers

What is the exponential form of 1,000?

<p>10^3</p> Signup and view all the answers

What is the simplified form of $(2^3)^2$?

<p>$2^6$ (A)</p> Signup and view all the answers

$(52)^3$ is greater than $(52) imes 3$.

<p>True (A)</p> Signup and view all the answers

What is the result of $(7^2)^{10}$?

<p>$7^{20}$</p> Signup and view all the answers

The expression $(a^m)^n$ can be simplified to $a^{______}$.

<p>$am imes n$</p> Signup and view all the answers

Match the following expressions with their simplified forms:

<p>(3^2)^4 = $3^8$ (2^4)^3 = $2^{12}$ (5^3)^2 = $5^6$ (6^1)^5 = $6^5$</p> Signup and view all the answers

What is the result of simplifying $2^3 imes 3^3$?

<p>$(2 imes 3)^3$ (B), $6^3$ (D)</p> Signup and view all the answers

The result of $(2^2)^{100}$ is equal to $2^{200}$.

<p>True (A)</p> Signup and view all the answers

If $a$ is a non-zero integer and $m$ and $n$ are whole numbers, what is the general formula for $(a^m)^n$?

<p>$a^{mn}$</p> Signup and view all the answers

The expression $3^5$ equals $3 imes 3 imes 3 imes 3 imes 3$.

<p>True (A)</p> Signup and view all the answers

According to the laws of exponents, $a^m imes a^n$ equals __________.

<p>a^(m+n)</p> Signup and view all the answers

Match the following exponent values with their results:

<p>$4^2$ = 16 $5^3$ = 125 $10^1$ = 10 $6^0$ = 1</p> Signup and view all the answers

What is the result of $35 imes 0$?

<p>0 (C)</p> Signup and view all the answers

Calculate the value of $2^4$.

<p>16</p> Signup and view all the answers

The expression $23 > 52$ is true.

<p>False (B)</p> Signup and view all the answers

What is the result of simplifying $32 × 34 × 38$?

<p>$3^{14}$ (B)</p> Signup and view all the answers

The expression $7^0$ is equal to 7.

<p>False (B)</p> Signup and view all the answers

Simplify $615 ÷ 610$ and express in exponential form.

<p>5</p> Signup and view all the answers

The result of $a^3 × a^2$ is $a^{______}$.

<p>5</p> Signup and view all the answers

What is the simplified form of $(2^3 × 4) ÷ 3 × 32$?

<p>$8^{2}$ (B)</p> Signup and view all the answers

The expression $10 × 10^{11} = 100^{11}$.

<p>False (B)</p> Signup and view all the answers

Match the following expressions with their exponential results:

<p>$5^2 ÷ 5^4$ = $5^{-2}$ $3^3 × 3^2$ = $3^5$ $a^4 × a^4$ = $a^8$ $2^0$ = $1$</p> Signup and view all the answers

What is the simplified form of the expression $8t ÷ 82$?

<p>1</p> Signup and view all the answers

Express the number 120719 in standard form.

<p>1.20719 × 10^5</p> Signup and view all the answers

The distance from the Sun to Earth is less than 1.5 × 10^11 m.

<p>False (B)</p> Signup and view all the answers

What is the standard form of the number 3,430,000?

<p>3.43 × 10^6 (A)</p> Signup and view all the answers

The standard form of the number 70,040,000,000 is __________.

<p>7.004 × 10^10</p> Signup and view all the answers

Match the following distances with their standard forms:

<p>Distance to Moon = 3.84 × 10^8 m Speed of light = 3.0 × 10^8 m/s Diameter of Earth = 1.2756 × 10^7 m Diameter of Sun = 1.4 × 10^9 m</p> Signup and view all the answers

Which of the following numbers is written in expanded form as 4 × 10^5 + 5 × 10^3 + 3 × 10^2 + 2 × 10^0?

<p>445320 (C)</p> Signup and view all the answers

In the number 5.9853 × 10^3, the exponent indicates that the decimal point is moved three places to the right.

<p>True (A)</p> Signup and view all the answers

Convert the number 30,000,000,000,000 to standard form.

<p>3.0 × 10^13</p> Signup and view all the answers

Flashcards

Exponents

A way to write large numbers in a shorter form using repeated multiplication.

Base

The number that is multiplied repeatedly

Exponent (Power)

The number that tells how many times the base is multiplied by itself.

Exponential Form

A shorter way to write a number using a base and an exponent.

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10 to the power of 4

10 multiplied by itself four times (10 x 10 x 10 x 10).

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Exponential form of 1000

10 multiplied by itself three times (10 x 10 x 10).

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Base 10

Numbers that have 10 as their base, meaning the number formed by repeatedly multiplying 10 by itself.

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Special names of powers

Some powers (especially of 10) have specific names (e.g., 10² is called 'ten squared', 10³ is called 'ten cubed').

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Expanding a³ b²

Expanding a³ b² means expressing the term as a product of its factors, (a × a × a) × (b × b) = a × a × a × b × b.

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Comparing a³ b² and a² b³

a³ b² and a² b³ are different because the powers of 'a' and 'b' are different. The order of factors does not affect the terms

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a³ b² = b² a³

When the powers of the variables remain the same, the order of terms does not change the final product.

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Order of Factors

The order in which factors are multiplied does not change the final result, as long as the powers remain the same.

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Difference in a³ b² and a² b³

The terms are different because the exponents of the variables differ.

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2^0

Any number raised to the power of zero is equal to 1

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3^0

Any number raised to the power of zero is equal to 1

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Exponent Rules: 3⁵ ÷ 3⁵

Resulting in 3 to the power of (5 – 5) equals 3 to the power of 0, which equals 1.

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7^0

Any number raised to the power of zero is equal to 1

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Power of Zero

Any nonzero number raised to the power of zero equals one.

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Example Finding 2^0

Following the pattern of 2^1 = 2, 2^2 = 4, 2^3 = 8,...2^0 = 1.

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3^5/3^5

3^5 divided by 3^5 gives 3 raised to the power of 5-5 which equals 3^0 is equal to 1

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Exponent rule for multiplying with the same exponent

When multiplying terms with the same base but different exponents, keep the base and add the exponents.

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(a^m)^n

When a power of a term is raised to a power, multiply the exponents.

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(6^2)^4

Applying the exponent rule, (a^m)^n = a^(m*n). Here this means, that the answer is 6^8.

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(2^2)^100

Applying the exponent rule, (a^m)^n = a^(m*n). Here this implies, the answer is 2^200

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(7^50)^2

Applying the exponent rule, (a^m)^n = a^(m*n). Here this implies the answer is 7^100

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(5^2)^3

Calculating this means raising the entire term 5^2 to the power of 3. (5^2)^3 = 5^6

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(5^2)*3

In this case, you first calculate 5^2 (which is 25) then Multiply by 3 gives 75.

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2^3 * 3^3

These two expressions have different bases, but multiplying them with the same exponent means multiplying the resultant number with itself 3 times, therefore, the answer will be 2 * 2* 2 * 3 3 3 =8 * 27.

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Standard Form

A way to express very large or very small numbers using powers of 10, making them easier to read and write.

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Standard Form Example

Express 5,000 in standard form.

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Distance to the Moon

What is the distance between Earth and the Moon in standard form?

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Speed of Light

What is the speed of light in vacuum in standard form?

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Diameter of the Earth

What is the diameter of the Earth in standard form?

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Standard Form for Large Numbers

Why is standard form useful for expressing large numbers?

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Simplify 32 × 34 × 38

Using the rule of exponents, multiply the numbers with the same base by adding their exponents. The result will be in exponential form.

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Simplify 615 ÷ 610

Divide numbers with the same base by subtracting the exponents. The result will be in exponential form.

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Simplify a3 × a2

To multiply the same bases, sum the exponents: am × an = am+n.

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Simplify 7x × 72

The rule for multiplying exponential terms with the same base is to add the exponents. In this situation, assume x is a variable.

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Simplify 52 ÷ 53

Divide the terms with like bases by subtracting the exponents. The result is in exponential form.

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Simplify 25 × 55

When multiplying exponential expressions with different bases, you multiply the base numbers and add the exponents. The result will be a product in exponential form.

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Simplify (34)3

When raising a power to a power, multiply the exponents. The result is in exponential form.

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Simplify 220÷215 ×23

Use the associative property to group the terms with the same base. Apply the rules for multiplying and dividing terms with the same base (add exponents for multiplication and subtract for division) to obtain the result in exponential form.

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Study Notes

Exponents and Powers

  • Exponents provide a shorter way to write large numbers
  • Large numbers are difficult to read and compare
  • Exponents are used to represent repeated multiplication
  • The number being multiplied is called the base
  • The number of times the base is multiplied is called the exponent
  • For example, 10,000 = 10 × 10 × 10 × 10 = 10⁴ (10 to the fourth power)

Exponential Notation

  • Large numbers can be written concisely using exponents
  • The base is repeatedly multiplied
  • The exponent indicates the number of times the base is used in multiplication
  • For example, 1,000 = 10 x 10 x 10 = 103 (base 10, exponent 3)

Laws of Exponents

  • Multiplying powers with the same base: Add the exponents
    • For example, 2² x 2³ = 2^(2+3) = 2⁵
  • Dividing powers with the same base: Subtract the exponents
    • For example, 2⁵ / 2² = 2^(5-2) = 2³
  • Power of a power: Multiply the exponents
    • For example, (2²)³ = 2^(2x3) = 2⁶
  • Multiplying powers with the same exponent: Multiply the bases
    • For example, 2³ x 3³ = (2x3)³ = 6³
  • Dividing powers with the same exponent: Divide the bases
    • For example, 6³ / 2³ = (6/2)³ = 3³
  • Zero exponent: Any non-zero number to the power of zero is equal to 1
    • a⁰ = 1 (e.g., 5⁰ = 1, 10⁰ = 1)

Standard Form of Large Numbers

  • Write numbers in the form a × 10ⁿ, where 1 ≤ a < 10 and n is an integer
  • Used to represent extremely large or small numbers concisely
  • For example, 300,000,000 = 3 × 10⁸

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Description

Test your understanding of algebraic expressions and exponents with this quiz. Determine equivalence, simplify expressions, and convert between standard and exponential forms. This quiz is perfect for students studying algebra concepts and looking to strengthen their skills.

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