Indiana 8 - 2020 Edition

1.05 Cube roots of perfect cubes

Lesson

Recall that raising a number to the power of three is often called "cubing" a number. For example, the expression $x^3$`x`3 has the following meanings:

$x^3$x3 |
a number $x$x raised to the power of three |
$x$x cubed |

Just as the square of a number relates to the area of a square, cubing a number relates to the volume of a cube.

Let's look at a table of the first ten perfect cubes. A perfect cube is a number that can be expressed as the cube of an integer. For example, the number $8$8 is a perfect cube because it can be expressed as $2\times2\times2$2×2×2 or $2^3$23.

$1$1 | $=$= | $1\times1\times1$1×1×1 | $=$= | $1^3$13 |

$8$8 | $=$= | $2\times2\times2$2×2×2 | $=$= | $2^3$23 |

$27$27 | $=$= | $3\times3\times3$3×3×3 | $=$= | $3^3$33 |

$64$64 | $=$= | $4\times4\times4$4×4×4 | $=$= | $4^3$43 |

$125$125 | $=$= | $5\times5\times5$5×5×5 | $=$= | $5^3$53 |

$216$216 | $=$= | $6\times6\times6$6×6×6 | $=$= | $6^3$63 |

$343$343 | $=$= | $7\times7\times7$7×7×7 | $=$= | $7^3$73 |

$512$512 | $=$= | $8\times8\times8$8×8×8 | $=$= | $8^3$83 |

$729$729 | $=$= | $9\times9\times9$9×9×9 | $=$= | $9^3$93 |

$1000$1000 | $=$= | $10\times10\times10$10×10×10 | $=$= | $10^3$103 |

If we are asked to find the cube root of a value, we are being asked, "What number multiplied by itself three times would give this value?"

We might also see the cube root symbol written with a number inside it, for example, $\sqrt[3]{125}$^{3}√125 represents the cube root of $125$125 which is equivalent to $5$5 because $5\times5\times5=125$5×5×5=125.

**Evaluate** $\sqrt[3]{64}$^{3}√64.

**Think: **We should read $\sqrt[3]{64}$^{3}√64 as "the cube root of $64$64".

This is the number multiplied by itself three times to make $64$64.

We know that $64=4\times4\times4$64=4×4×4.

**Do: **This means the cube root of $64$64 is $4$4, so $\sqrt[3]{64}=4$^{3}√64=4.

Evaluate $\sqrt[3]{27}$^{3}√27

Solve $x^3=64$`x`3=64.

Use square root symbols to represent solutions to equations of the form x^2 = p, where p is a positive rational number.