SME
Grade 8 Math · The Number System
Square roots, cube roots and irrational numbers
Use this for “Between which two whole numbers is √10?” and the side of a square from its area.
Diagram
A square of area 10 has side √10 ≈ 3.16
9 < 10 < 16, so √10 is between 3 and 4.
This is an example. Type your own values to start a new problem.
s
=
A
lo
Whole number at or below √A
worked out (not typed)
hi
Whole number at or above √A
worked out (not typed)
Formulas
s
=
√
A
(side
of
the
square
=
√area
of
the
square)
√
10
=
√
10
lo
=
whole
number
at
or
below
the
square
root
of
A
(whole
number
at
or
below
√A
=
whole
number
at
or
below
the
square
root
of
area
of
the
square)
3 = whole number at or below the square root of 10
hi
=
whole
number
at
or
above
the
square
root
of
A
(whole
number
at
or
above
√A
=
whole
number
at
or
above
the
square
root
of
area
of
the
square)
4 = whole number at or above the square root of 10
Assumptions
•
A square of area A has side √A, because side × side = area.
•
If A is a perfect square, √A is a whole number.
•
If not, √A is irrational: it lies between two whole numbers and its decimal never ends or repeats.
•
To place √A, find the perfect squares on either side.
Step-by-step
WE KNOW
A = 10 (area of the square)
FIND
side of the square (s), whole number at or below √A (lo), whole number at or above √A (hi)
Step 1 · Find side of the square (s)
Use s = √A (side of the square = √area of the square)
The side is the number that, times itself, makes the area.
s
=
√
10
To
the
tenths:
3.1
2
=
9.61
and
3.2
2
=
10.24
To
the
hundredths:
3.16
2
=
9.9856
and
3.17
2
=
10.0489
s
=
√
10
≈
3.1623
Step 2 · Find whole number at or below √A (lo)
Use lo = whole number at or below the square root of A (whole number at or below √A = whole number at or below the square root of area of the square)
Find
the
biggest
perfect
square
that
is
not
more
than
A
.
lo = whole number at or below the square root of 10
3
2
=
9,
and
9
≤
10
4
2
=
16,
which
is
more
than
10
lo = 3
Step 3 · Find whole number at or above √A (hi)
Use hi = whole number at or above the square root of A (whole number at or above √A = whole number at or above the square root of area of the square)
Find
the
smallest
perfect
square
that
is
not
less
than
A
.
hi = whole number at or above the square root of 10
4
2
=
16,
and
10
≤
16
hi = 4
Check
Put the numbers back in. Both sides should match.
3.1623
≈
√
10
✓
3 = whole number at or below the square root of 10 ✓
4 = whole number at or above the square root of 10 ✓
Topic check
See what you know: 5 questions on this topic
New questions each time. Pass to mark it complete.
Refresh
More problem types
Cube roots as edges
Use this for “A cube has volume 1,000 cm³. What is its edge?”
A repeating decimal as a fraction
Use this for “Write 0.363636… as a fraction.”
Rational or irrational?
Use this for “Which of these numbers are irrational: −11/4, 0.2468, √41, −√81?”
Does the decimal end or repeat?
Use this for “Which fractions have decimals that end: 3/8, 1/3, 7/20?”