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3.6 Chapter summary

  • When writing exponents, we use a small superscript to show how many times a number is multiplied by itself. For example, in \(2^{\mathbf{5}} = 2 \times 2 \times 2 \times 2 \times 2\), the number \(2\) is multiplied by itself \(\mathbf{5}\) times. We read this as “\(2\) to the power of \(\mathbf{5}\)”.
  • When the bases are different and the powers are the same, we can compare the exponents by comparing only the bases. So, \(3^2 < 5^2\) because \(3<5\).
  • When the bases are the same and the powers are different, we can compare the exponents by comparing only the powers. So, \(3^3>3^2\) because \(3>2\).
  • When the base is a negative number and the exponent is an even number, the answer is positive: \(( - 5)^{2} = 25\)
  • When the base is a negative number and the exponent is an odd number, the answer is negative: \(( - 5)^{3} = - 125\)
  • In scientific notation, a number is expressed in two parts: a number between \(1\) and \(10\) multiplied by a power of \(10\). The exponent must always be an integer. \(\text{1,36} \times 10^8\) is called the scientific notation for \(136\) \(000\) \(000\).
  • To find the square root of a number, we ask the question: Which number was multiplied by itself to get a square? The answer to this question is written as \(\sqrt{16} = 4\).
  • To find the cube root of a number we ask the question: Which number was multiplied by itself and again by itself to get a cube? The answer to this question is written as \(\sqrt[3]{64} = 4\).
  • The exponential form \(a^{b}\) is read as “\(a\) to the power of \(b\)”, where \(a\) is called the base and \(b\) is called the exponent or index. The exponent indicates the number of times a factor is multiplied.
  • When you multiply two or more powers that have the same base, the answer has the same base, but its exponent is equal to the sum of the exponents of the numbers you are multiplying. We can express this algebraically as \(a^m \times a^n = a^{(m+n)}\), where \(m\) and \(n\) are natural numbers and \(a\) is not zero (\(m\) \(\mathbb{\in N}\), \(n\) \(\mathbb{\in N}\) and \(a \neq 0\)).
  • Any number raised to the power \(1\) is equal to the number itself: \(a^{1}=a\).
  • The number \(1\) raised to any power is equal to \(1\): \(1^{a}=1\).
  • Any number raised to the power \(0\) is equal to \(1\): \(a^{0}=1\).
  • When you divide powers with the same base, the answer has the same base, but its exponent is equal to the difference of the exponents of the numbers you are dividing.
    We can express this algebraically as \(a^{m}\) \(\div\) \(a^{n}\) \(=\) \(a^{m-n}\); \(m\) \(\mathbb{\in N}\), \(n\) \(\mathbb{\in N}\) and \(a \neq 0\).
  • When a base is raised to more than one power, the powers are multiplied.
    We can express this algebraically as \(\left( a^{m} \right)^{n}\) \(=\) \(a^{m \times n}\); \(m\) \(\mathbb{\in N}\), \(n\) \(\mathbb{\in N}\) and \(a \neq 0\).
  • A product raised to a power is the product of the factors, with each raised to the same power. Using algebraic notation, we write \((a \times b)^{m} = a^{m} \times b^{m}\), \(m\) \(\mathbb{\in N}\), \(a≠0\) and \(b≠0\).